Exotic Spheres Johnny Nicholson

Part III Essay submitted to the University of Cambridge 2016/2017

Contents

1 Introduction 1

2 Characteristic Classes 4 2.1 Fibre Bundles ...... 4 2.2 Chern Classes ...... 6 2.3 Pontryagin Classes ...... 8

3 Milnor’s Exotic Spheres 11 3.1 Morse Theory ...... 11 3.2 Milnor’s λ Invariant ...... 13 3.3 Higher Dimensions ...... 16

4 Cobordism 18 4.1 Hirzebruch’s Signature Theorem ...... 20 4.2 h-Cobordism Theorem ...... 24

5 Plumbing 27 5.1 Plumbing Disc Bundles ...... 27 5.2 Examples of Exotic Spheres ...... 30

6 Brieskorn Varieties 32 6.1 The Alexander Polynomial of K ...... 34 6.2 Examples of Exotic Spheres ...... 36

7 Classification of Exotic Spheres 38

8 References 42 1 Introduction

A smooth n-manifold is called an exotic sphere if it is homeomorphic but not di↵eomorphic to Sn. Until 1953, when John Milnor discovered the first example of an exotic sphere in dimension 7, it was widely believed that no such object existed. This was firstly because of the subtleties underlying the definitions involved, though also due to the way in which dimension was believed to a↵ect problems in topology. We begin by illustrating some of these subtleties and giving an overview of what was known at the time of this discovery. 1. Firstly, recall that the definition of a smooth manifold M is suciently generous that it allows us to choose any (possibly pathological) chart to define the smooth structure so long as it is smoothly compatible with the other charts chosen. For example (R, ') is a smooth manifold for any homeomorphism ' : R R. ! This is rectified by choosing a similarly generous notion of equivalence. Namely a di↵eomorphism between smooth manfolds M and N can be any homeomorphism that induces smooth maps on the coordinate patches. For example (R, ')and(R, ) 1 are di↵eomorphic via the homeomorphism ' : R R. ! We will use smooth structure (on M)torefertoanequivalenceclassofsmooth manifolds homeomorphic to M up to di↵eomorphism. 2. Secondly we remark that, before 1953, little was known about smooth structures on manifolds except in very small dimensions where direct proofs are often tractable. For example, to show S1 has a unique smooth structure, note that a homeomorphism 1 1 1 ' : U '(U) R for U S open gives a local di↵eomorphism ' : (a, b) S ! ✓ ✓ 1 ! for some a, b R . The image of ' is open and, by stitching it together 2 [ {1} 1 with local di↵eomorphisms around the boundary points of ' ((a, b)), we can also 1 1 assume the image of ' is closed. This shows that ' is surjective and must have (a, b)=R. It can then be checked that it descends to a di↵eomorphism R/Z S1. ! With more diculty, it had also been shown that S2 and S3 had a unique smooth structure1 and it was thought that the higher dimensional cases would be similar. It was therefore considered very surprising when exotic spheres were shown to exist, and later even more surprising that they could be constructed in so many di↵erent ways and even classified in any given dimension. It was also shown in 1960 that not every topological manifold admitted a smooth structure (see [5]). Clearly these definitions have a lot more in them than was initially thought, and it is exploring this that will be the starting point for this exposition. Suppose we wanted to prove that exotic n-spheres do not exist. We might attempt to use that every homeomorphism f : M Sn can be uniformly approximated by a smooth map M Sn though, whilst we get! smooth maps M Sn and Sn M,wehaveno way of amalgamating! these approximations to give a di↵!eomorphism. Whilst! this may be extremely counterintuitive, the reason for this comes from the fact that homeomorphisms can be far more pathological than our intuition usually suggests. On the other hand, suppose we wanted to prove that exotic n-spheres do exist. Since no progress was (or ever could have been) made for n 3, we would be interested in any result that is unique to higher dimensions. For example, in 1962 Stephen Smale proved

1In fact, it was proven later that every smooth n-manifold has a unique smooth structure for n 3.  the h-cobordism theorem which implied the Poincar´econjecture2 in dimensions n 5. These ideas can be applied to the following construction which arose in the 1960s, known as twisting. Consider the smooth n-manifold ⌃ = Dn Dn, f tf n 1 n 1 where f : S S is an orientation-preserving di↵eomorphism, and assume n 5. ! Since f must have degree 1, it is homotopic to the identity and so ⌃f is homotopic to n n n n S = D id D . Hence ⌃f is homeomorphic to S by the Poincar´econjecture for n 5. Furthermore,[ the h-cobordism theorem can be used to show that every smooth n-manifold homeomorphic to Sn can be represented in this form. It is then not dicult to see that n 1 n 1 smoothly isotopic di↵eomorphisms f : S S induce di↵eomorphic manifolds ⌃f . In fact, it was shown by Jean Cerf in 1970 that! the two are in correspondence.

Figure 1: Discs Dn Dn (left) being attached by f to form ⌃ (right). t f Whilst it may seem a good approach to study these smooth isotopy classes instead, explicit examples of di↵eomorphisms not smoothly isotopic to the identity are dicult to work with3. Instead, what we should take away from this viewpoint is an understanding of how smooth structures can be inequivalent: whilst f can be “shrunken to a point” to give a homeomorphism to Sn, the derivatives of f may not behave nearly so well. The primary focus of this essay will be to give an overview of the various constructions of exotic spheres as well as the techniques used for proving firstly that they are homeomorphic to Sn and secondly that they are not di↵eomorphic to Sn. The first of these tasks usually amounts to a straightforward computation of the (co)homology groups and fundamental group of a space, with the heavy lifting being performed by the Poincar´econjecture for n 5, though an alternate approach using Morse Theory will also be considered for completeness. With this is mind, the main obstacle lies is in developing invariants that are able to distinguish smooth structure. As a result, we will consider constructions which lend themselves particularly well to such an analysis. The exotic spheres discovered by Milnor were constructed as S3 bundles over S4. In this case, the invariant we will develop that distinguish smooth structure will come from invariants of vector bundles, namely characteristic classes. We thus begin by giving an introduction to fibre bundles and characteristic classes before expanding the details of Milnor’s original paper. We then show how these ideas can only be made to work in dimensions 7 and 15. We next take a detour to consider the theory of cobordism, an equivalence relation whereby two compact manifolds are equivalent if their disjoint union is the boundary

2This states that every oriented compact simply-conncted smooth n-manifold M with the homology of Sn is homeomorphic to Sn. In particular, every such homotopy n-sphere is homeomorphic to Sn 3Such maps can however be written down using already known exotic spheres. See [8] for an example.

2 of a compact manifold of one dimension higher. Here we prove Hirzebruch’s signature theorem, the key ingredient in defining the invariant we previously used, and briefly discuss the h-cobordism theorem. In search of more exotic spheres, and armed with various machinary, we then consider two further constructions known as Plumbing and Brieskorn varieties. These constructions will be restricted to the odd-dimensional cases and, common to both constructions of a suitible closed (2n 1)-manifold M, will be a bounding 2n-manifold B. In fact, we can choose our constructions so that all the Msweconsiderwillbe(n 2)-connected and 2n 1 the Bswillbe(n 1)-connected. Thus showing M is homeomorphic to S amounts to showing just one further connectivity condition, a condition we can characterise in a number of ways. We also find many interesting links to Lie groups and knot theory. We end by discussing the classification of exotic spheres of a given dimension, showing that this is related to the problem in homotopy theory of computing the stable homotopy groups of spheres. The key step is to use the h-cobordism to allow us to swap di↵eomor- phism for the much easier to work with condition of being h-cobordant. The computations then rely on showing that the exotic n-spheres form a group ⇥n under connected sum and identifying the subgroup bPn+1 of manifolds which bound manifolds with trivial tangent bundle, the bounded parallelisable manifolds.

3 2 Characteristic Classes

In this chapter, we start by developing the basic theory of fibre bundles before describing a correspondence between fibre bundles over spheres and higher homotopy groups that we will use to construct exotic spheres in the following chapter. We next use the Euler class to construct further characteristic classes and give an overview of the various techniques we will need for computing them. Here and elsewhere in the essay, we will rely on basic facts from Homotopy Theory. We will assume without further mention the definition of higher homotopy groups and the basic theory of vector bundles. Other facts will be stated briefly without proof, with the primary reference being the fourth chapter of [13]. 2.1 Fibre Bundles It is well known that any (real) vector bundle Rn , E X has an associated sphere n 1 n ! ! bundle S , S(E) X and disc bundle D , D(E) X by operating on the fibres. All of these bundles! can! be considered under the! following! heading: Definition (Fibre bundle). A fibre bundle structure on a space E is a pair of spaces F and X equipped with a projection map ⇡ : E X such that there is an open cover 1 ! Ui of X and homeomorphisms hi : ⇡ (Ui) Ui F coinciding with ⇡ in the first coordinate.{ } We write: ! ⇥ F E ⇡ X. 1 In particular, this means that each fibre F = ⇡ (x)mapshomeomorphicallyto x F . x { }⇥ We call E the total space, X the base space, F the fibre and hi the (local) trivialisations. This also generalises covering spaces which are fibre bundles with discrete fibres. For n n example, the covering map Sn RP gives a fibre bundle 1 , Sn RP . Note that, when n =1,thisspecialisestothespherebundle! S0 , {±S1 } !S1. We! may wonder if there are any other fibre bundles where all three spaces are! spheres:!

n+1 n Example. (i) The quotient map (C )⇥ CP has fibres C⇥ and induces a fibre n ! bundle S1 , S2n+1 CP . When n =1thisgives: S1 , S3 S2. ! ! ! ! n (ii) If H is the algebra of quaternions, we can similarly define HP = Hn+1/ where 4 ⇠ (q0,...,qn) (cq0, . . . , cqn)foranyc H⇥. Since H = R as spaces, we get a fibre ⇠ n 12 bundle S3 , S4n+3 HP . Now HP = [q, 1] : q H [1, 0] = R4 = S4 ! ! { 2 }[{ } ⇠ [{1} ⇠ is the compactification of R4. Hence when n =1thisgives: S3 , S7 S4. ! ! Remark. The only further example of this construction (and the only other real finite- dimensional division algebra) is the octonionic algebra O which gives S7 , S15 S8. In fact, it can be shown that these are the only fibre bundles where F , E and!X are! spheres. Note that, when defining vector bundles, each fibre comes equipped with the structure of avectorspacethathastoberespectedbythetrivialisations.Moregenerallywehave: Definition (Structure group). Let F , E X be fibre bundle and G agroupacting ! ! on F by homeomorphisms such that any Ui, Uj in the trivialising open cover has

1 h h : (U U ) F (U U ) F, (x, y) (x, g (x)y) i j i \ j ⇥ ! i \ j ⇥ 7! ij for some continuous gij : Ui Uj G. We call G the structure group and such a fibre bundle a G-bundle. \ !

4 Example. For a fibre bundle Rn , E X, giving a description of how GL(n, R)acts on the fibres is equivalent to giving! a vector! space structure on each fibre. Hence a real vector bundle is precisely a fibre bundle with fibre Rn and structure group GL(n, R). Furthermore, note that giving an orientation on the vector bundle corresponds to a con- tinuous choice of orientation on the fibres. This is equivalent to describing an action by SL(n, R)andsoorientedvectorbundlesarefibrebundleswithstructuregroupSL(n, R). n 1 Finally, this action on the correponding sphere bundle S , S(E) X then corre- ! ! sponds to O(n, R)andtoSO(n, R)fororientedvectorbundles. We can define a morphism of fibre bundles exactly as for vector bundles and can also make similar definitions in the category of smooth manifolds: if all spaces above are smooth manifolds and all maps are smooth, then we call such a fibre bundle a smooth fibre bundle. In general, classifying fibre bundles up to isomorphism is a dicult task. However, in the case that the base space is a sphere, such fibre bundles are characterised by the following construction known as clutching. First observe that, given a topological group G of di↵eomorphisms of a space F , we can construct a class of smooth fibre bundles over Sn with fibre F and structure group G from n 1 amapf : S G (the clutching map)asfollows. ! n 1. First let UN and US be S without the north and south poles respectively and note that f induces a map f¯ : U U G by projecting to the equator. N \ S ! 2. Now let E =(U F ) (U F )/ , where (u, x) (u, f¯(u)x). f N ⇥ t S ⇥ ⇠ ⇠ 3. Then F , E Sn is a fibre bundle with projection onto the first coordinate. ! f ! Remark. This may remind us of the twisting construction from the introduction, though the two are very di↵erent: whilst here the gluing is trivial along the zero section, the gluing done in the twisting construction can be non-trivial everywhere. Indeed all such smooth fibre bundles arise in this way (for a proof, see [14]): Theorem. Every smooth fibre bundle over Sn with fibre F and structure group G is isomorphic to one obtained by the construction above, and Ef ,Eg are isomorphic i↵ n 1 f,g : S G are homotopic, i.e. there is a correspondence ! n n ⇡n 1(G) FibF (S )= G-bundles over S . ! { }

Hence we can use knowledge of ⇡n 1(G)togiveusasupplyofsmoothfibrebundlesofa particular form. We now show how this can be used to construct S3-bundles over S4. Example. We showed above that ensuring transition functions induce isometries on S3 is the same as requiring that fibre bundle has structure group SO(4). Hence constructing 3 4 S -bundles over S amounts to finding an explicit form for ⇡3(SO(4)). Each SO(4) acts on by rotations on S3, which we take to be the unit ball in the quaternionic2 space generated by 1, i, j, k . If SO(3) is the subgroup of rotations in the 1 { } i, j, k -plane, then (1) SO(3) since it fixes 1. Hence we have a homeomorphism { } 2 SO(4) = S3 SO(3) ⇠ ⇥ 1 given by ((1), (1) )andwithinverse(u, ) (v u (v)). Now consider 7! 7! 7! 3 1 ⇢ : S SO(3),u (v uvu ), ! 7! 7! 5 where multiplication is quaternionic. To show the image is in SO(3), observe that it must be in O(3) since it is an R-linear, norm-preserving map fixing the identity. It contains the identity and is connected, since S3 is connected, and so lies in SO(3). We claim this is a two-sheeting covering map. This can be shown by, for example, noting 3 that an isomorphism SO(3) RP can be constructed that makes ⇢ into the quotient 3 3 ! map from S to RP . We can now get an explicit form for ⇡3(SO(4)) as follows. 3 1. First note that ⇡3(S )=Z and its elements can be represented in the form : S3 S3,u ui i ! 7! for i Z. This is proven using the Hurewicz theorem which states that the first non trivial2 H (S3)fork 1coincideswith(theabelianisationof)⇡ (S3). k k 2. Next, we note that ⇡3(SO(3)) = Z and its elements can be represented in the form ⇠ 3 ⇢ j for j Z. This is proven using that covering maps ⇢ : S SO(3) induce isomorphisms 2 on higher homotopy groups. ! 3. Combining with the homeomorphism S3 SO(3) SO(4) established above, we have ⇥ ! S3 S3 SO(3) SO(4) ⇥ i j i+j j u (u , ⇢(u )) (v u vu ) 7! i j j j j with the last term coming from v u ⇢(u )(v)bynotingthat⇢(u )(v)=u vu . 7! There are no further elements since4 ⇡ (X Y ) = ⇡ (X) ⇡ (Y )foranyX, Y . n ⇥ ⇠ n ⇥ n 2 Hence ⇡3(SO(4)) ⇠= Z and its elements can be represented in the form : u (v uivuj), ij 7! 7! where we substitute (i, j) (i + j, j). 7! Write E to refer to the smooth fibre bundle S3 , E S4 corresponding to ij ! ij ! ij 2 ⇡3(SO(4)). We will return to this in the following chapter to show that the Eij are homeomorphic to S7 when i + j = 1andthatmanyareinfactexoticspheres. ± 2.2 Chern Classes Throughout this section, we will use Z coecients and let ⇠ =(E,X,⇡)denotead- dimensional vector bundle with total space E, base space X and projection ⇡. We also use E# to denote E s (X), where s : X E is the zero section. \ 0 0 ! In the case where ⇠ admits a Z-orientation "x x X , recall that there is a unique class d # { } 2 u⇠ H (E,E )(theThom class) which restricts to "x on each fibre. The Euler class e(⇠)2 Hd(X)isthendefinedbytheimageofu under the composition 2 ⇠ s Hd(E,E#) Hd(E) 0⇤ Hd(X) . ⇠

This has the property that, if ⇠0 =(E0,X0, ⇡0)isanotherorientedd-dimensional real vector bundle and (F, f) : ⇠ ⇠0 is an orientation-preserving bundle morphism, then ! f ⇤e(⇠0)=e(⇠). We say that e satisfies the naturality condition.

4 n This is an immediate consequence of the fact that we can combine homotopies hX : S I X and h : Sn I Y to give homotopies (h ,h ) : Sn I X Y . ⇥ ! Y ⇥ ! X Y ⇥ ! ⇥ 6 That is, e(⇠) is an invariant for the category of oriented real vector bundles. We call an invariant of this form for (some set of) vector bundles a characteristic class. We will primarily be concerned with the following property of the Euler class: Definition (Gysin sequence). The long exact sequence for the pair (E,E#)gives:

Hi+d(E,E#) Hi+d(E) Hi+d(E#) Hi+d+1(E,E#) ⇠ ⇠ ⇡⇤ s0⇤ ⇡0⇤ ^e(⇠) Hi(X) · Hi+d(X) Hi+1(X) where is the Thom isomorphism, given by cupping with the Thom class. The bottom sequence, with maps such that the diagram commutes, is the Gysin sequence. Since the top row is exact, the Gysin sequence must be also. This shows:

i i # Proposition. ⇡⇤ : H (X) H (E )isanisomorphismforideterminant splits GL(n, R)intotwoconnectedcomponents. Hence we immediately get a top-dimensional invariant of complex vector bundles, namely 2d e(⇠R) H (X). To find more invariants, observe that the above proposition gives a way to relate2 cohomology classes of vector bundles over X to those of vector bundles over E#. Fortunately, we have a natural way to construct a bundle over E# as follows: # # 1. For each v Ex E , let Ev = Ex/Cv. 2 ✓ # 2. Now let E = v E# Ev with the natural projection map ⇡ : E E . 2 e ! 3. Then ⇠# =(E,ES #, ⇡)isa(d 1)-dimensional complex vector bundle. e e e # e 2(d 1) # Since this is one dimension lower, it has an invariant e(⇠R ) H (E )whichindeed e e 1 # 2(d 1) 2 gives us an invariant (⇡0⇤) e(⇠R ) H (X)oftheoriginalvectorbundle.Bycon- sidering a sequence ⇠, ⇠#, (⇠#)#,...2 of vector bundles, we get a family of invariants for ⇠: Definition. We define the ith Chern class c (⇠) H2i(X)inductivelyond by: i 2

e(⇠R),i= d. ci(⇠)= 1 # ((⇡0⇤) ci(⇠ ),i

It is often useful to package all this information as a single element in the graded ring H⇤(X). In particular, the total Chern class is defined as

c(⇠)=1+c (⇠)+c (⇠)+ + c (⇠) H⇤(X). 1 2 ··· d 2 In certain cases, as in the following example, computing the Chern classes is simply a consequence of noting that particular cohomology groups of the base space vanish.

7 Example. Recall that, similarly to R and C, we can define the tautological (line) bundle over the quaternions H

n n H , n = (x, v) HP Hn+1 : [v]=x HP ! H { 2 ⇥ } ! which we view an a 2-dimensional complex vector bundle by identifying H with C2.We 1 will consider the case n =1.WeremarkedearlierthatHP is di↵eomorphic to S4 and so 1 2 1 2 4 1 1 1 c1( ) H (HP )=H (S )=0.Ife = e( ), then c2( )=e and so c( )=1+e. H 2 H H H n Remark. The result is identical for n>1sinceH2(HP )=0.Toshowthis,weusethe n n+1 n n Gysin sequence to compute H⇤(HP )=Z[e]/(e )justasforRP and CP . In general, computing the Chern classes usually requires understanding how the vector bundle relates to known cases like the one just considered. We now state a few basic properties of Chern classes, the proofs of which are standard and can be found in [15].

Proposition. Let ⇠ =(E,X,⇡)and⇠0 =(E0,X0, ⇡0)becomplexvectorbundles.Then:

(i) If (F, f) : ⇠ ⇠0 is a bundle map, then f ⇤c (⇠0)=c (⇠). ! i i (ii) If "k : X Ck X is the trivial k-bundle, then c(⇠ "k)=c(⇠). ⇥ ! (iii) If ⇠ ⇠0 denotes the Whitney sum, then c(⇠ ⇠0)=c(⇠)c(⇠0). (iv) If ⇠ denotes the conjugate bundle, formed by composing each trivialisation with the i map that takes the complex conjugate on each copy of C, then ci(⇠)=( 1) ci(⇠). Example. To compute the Chern classes of the tangent bundle

n n Cn , T CP CP , ! ! n recall that CP = S2n+1/ where u u for all S1. Then ⇠ ⇠ 2 n T CP = [u, v] : (u, v) S2n+1 Cn+1,u v =0 , { 2 ⇥ · } where [u, v] is the equivalence class of (u, v) Cn+1 Cn+1 under (u, v) (u, v)forall 2 ⇥ ⇠ S1. We can write the trivial 1-bundle as "1 = [u, µu] : u S2n+1,µ C . Since u, v 2 n { 2 2 } are orthogonal in T CP ,wehave

n T CP "1 = [u, v + µu] : (u, v) S2n+1 Cn+1,µ C,u (v + µu)=Re(µ) ⇠ { 2 ⇥ 2 · } = [u, v + µu] : (u, v + µu) S2n+1 Cn+1 = [u, v] : (u, v) S2n+1 Cn+1 ⇠ { 2 ⇥ } ⇠ { 2 ⇥ } where the second isomorphism is since requiring that u v0 =Re(µ)forsomeµ C is no · 2 restriction at all. Since n = [u, v] : (u, v) S2n+1 C , this shows that C { 2 ⇥ } n T CP "1 = n n, ⇠ C ··· C n where there are n +1 terms in the sum. It is easy to see that c(C)=1+a where n 2 n n n a = e(C) H (CP ), and this is generator by the Gysin sequence for C , C CP . Hence we get2 ! ! n n c(T CP )=c(T CP "1)=c(n)n+1 =(1+a)n+1. C 2.3 Pontryagin Classes Recall that we found a characteristic class for oriented real vector bundles and then used it to construct an invariant for complex vector bundles. Interestingly we can now get an invariant for not-necessarily-oriented real vector bundles ⇠ as follows.

8 Definition. We define the ith Pontryagin class p (⇠) H4i(X)by: i 2 i pi(⇠)=( 1) c2i(⇠ C), ⌦R where ⇠ C is the complex vector bundle formed by tensoring each fibre with C. ⌦R

Remark. We only consider the even Chern classes since ⇠ R C = ⇠ R C implies that 2i+1 ⌦ ⇠ ⌦ c2i+1(⇠ C)=( 1) c2i+1(⇠ C)andso2c2i+1(⇠ C)=0. ⌦R ⌦R ⌦R We can similarly define the total Pontryagin class

p(⇠)=1+p (⇠)+p (⇠)+ p (⇠) H⇤(X), 1 2 ··· n 2 where n is maximal so that 4n d, the dimension of ⇠. Most of the basic properties of of Chern classes carry over immediately, with one small change:

Proposition. Let ⇠ =(E,X,⇡)and⇠0 =(E0,X0, ⇡0)becomplexvectorbundles.Then p(⇠ ⇠0)=p(⇠)p(⇠0) modulo 2-torsion elements. i Proof: The above remark shows that c(⇠ C) ( 1) pi(⇠)modulo2-torsion.Since ⌦R ⌘ (⇠ ⇠0) C = (⇠ C) (⇠0 C), the product formula for Chern classes gives ⌦R ⇠ ⌦R ⌦R P k i j ( 1) p (⇠ ⇠0) ( 1) p (⇠) ( 1) p (⇠0) k ⌘ i · j i j Xk X X k ( 1) p (⇠)p (⇠0)modulo2-torsion. ⌘ i j Xk i+Xj=k k We can then cancel the ( 1) sbynotingthisholdsateachgradeinH⇤(X). We now introduce a useful computational tool that allows us to compute Pontryagin classes from Chern classes. Proposition. Let ⇠ be a complex vector bundle. Then 1 p (⇠ )+p (⇠ ) =(1 c (⇠)+c (⇠) )(1 + c (⇠)+c (⇠)+ ). 1 R 2 R ··· 1 2 ··· 1 2 ··· Proof: This amounts to noting that ⇠ C is isomorphic to ⇠ ⇠, which can be checked on R ⌦R the fibres. It then follows directly from c(⇠R R C)=c(⇠)c(⇠)andthefactthatthe ⌦ k odd terms vanish completely: c2i+1(⇠R R C)= ( 1) ck(⇠)c2i+1 k(⇠)=0. ⌦ k Example. We can then use this to continue the examplesP we established before. 1 1 (i) For the line bundle H over HP , we now have: 1 p (1 )+p (1 ) =(1+e)2 =1+2e + e2, 1 H 2 H ··· where e = e(1 ). Hence p(1 )=1 2e + e2, i.e. p (1 )= 2e and p (1 )=e2. H H 1 H 2 H n (ii) For the tangent bundle of CP we now have: n n n n+1 2 n+1 1 p1(T CP )+p2(T CP ) =(1 a) (1 + a) =(1 a ) , ··· n n 2 n+1 n n+1 2i where a = e(C). Hence p(T CP )=(1+a ) , i.e. pi(T CP )= i a . Example. The n-sphere has tangent and normal bundle: n n+1 TuS = v R : u v =0 , ( Sn Rn+1 )u = Ru. { 2 · } V ✓ n n n n+1 for any u S . Now TS is the trivial bundle since TuS u = R . Since is also trivial,2 we get p(TSn)= Vp(TSn )=1,i.e. p (TSn)=0forall V i. V V i 9 We will now conclude this chapter by considering how the Pontryagin classes can be used to give invariants for smooth manifolds. Consider a compact connected oriented smooth 4i manifold M and the Pontryagin classes pi(TM) H (M) of its tangent bundle. Whilst previously we wanted to compare di↵erent vector2 bundles with the same base space, we now want to compare the tangent bundles of di↵erent manifolds M and N. To do this, we have to get around the issue that the Pontryagin classes lie inside H⇤(M)andH⇤(N) which do not necessarily have a canonical identification. The solution is to extract an integer-valued invariant, and this can be done as follows when M is 4d-dimensional.

First recall that we have a canonical element in H4d(X)givenbythefundamentalclass [M] (corresponding to an orientation on M). Taking the cap product of an element in H4d(M)with[M] then induces a map

4d H (M) H0(M) = Z ! ⇠ where we fix an identification with Z (this is of course one the Poincar´eduality maps). Furthermore, for any partition (i , ,i )ofd, we get an element of H4d(M)givenby 1 ··· r pi1 (TM) ^ ^ pir (TM) since this has grading 4i1 + +4ir =4d. This motivates the following··· definition. ··· Definition. Let M be a compact connected oriented manifold of dimension 4d and I = (i , ,i )apartitionofd. The Ith Pontryagin number is 1 ··· r

pI [M]=pi1 pir [M]=(pi1 (TM) pir (TM))[M] Z. ··· ··· 2 2n Example. Consider the compact connected 4n-dimensional smooth manifold CP . Note that this is orientable since, as before, its underlying real manifold has a canonical orien- tation. We showed that 2n 2n +1 2i pi(T CP )= a , i ✓ ◆ 2n 2n where a H2(CP )isagenerator,soweneedtocomputea2n[CP ]. To do this, note that 2 2n 2n 2n 2n 2n 2n e(T CP )[CP ]=c2n(T CP )[CP ]=(2n +1)a [CP ]. A standard fact5 about the Euler class, and the reason for its name, is that e(TM)[M]= (M)where is the Euler characteristic of M. We compute

2n 2n (CP )= ( 1)irank Hi(CP )=2n +1 i X 2n and so we must conclude that a2n[CP ] = 1. Hence

2n 2n +1 2n +1 pI [CP ]= . i ··· i ✓ 1 ◆ ✓ r ◆ Remark. We can similarly construct Chern numbers to get invariants for complex mani- folds, though will not consider them in this essay.

5For a proof, see chapter 11 of [15].

10 3 Milnor’s Exotic Spheres

The goal of this chapter will be to construct the exotic spheres discovered by Milnor in 1953, expanding the details of the arguments given in [1]. The spaces Milnor considered were the sphere bundles S3 , E S4 ! ij ! we constructed in the previous chapter, for each i, j Z. More specifically, let Mk be the total space E when i + j =1andi j = k. 2 ij 7 We will start by showing that the Mk are homeomorphic to S using the tools that were available to Milnor at the time, namely Morse theory6. We next use Pontryagin classes to develop an invariant that can distinguish smooth structure; this will rely heavily on Hirzebruch’s signature theorem, the proof of which we delay to the following chapter. This chapter then concludes by showing how these ideas can also be used to construct exotic 15-spheres but do not work in other dimensions. 3.1 Morse Theory The observation of Morse theory is that topological information about a manifold M can be inferred from knowledge of the existence of a smooth map f : M R with certain properties. In particular, we would like such functions to be of the following! form. Definition. Let M be a smooth compact manifold. We say a smooth map f : M R is a Morse function if every critical point is non-degenerate, i.e. if p M has! local 2 @f @2f coordinates x1, ,xn around p with @x =0foralli, then det @x @x =0. ··· i p i j p 6 ✓ ◆ This is well-defined since it can be shown that non-degeneracy of a critical points is a coordinate-independent phenomenon. We have the following key lemma. Lemma (Morse’s Lemma). Let M be a smooth compact n-manifold and f : M R a ! Morse function. If p M is a critical point, then there are local coordinates v1, ,vn around p such that 2 ···

f(v , ,v )=f(p) v2 v2 + v2 + + v2 1 ··· n 1 ··· k k+1 ··· n for some 1 k n 1whichwecalltheindex of f.   Intuitively this says that we can pick coordinates on a neighbourhood U around a critical point p so that each coordinate axis cuts through U in such a way that its image under f gives the shape of a “ x2-graph”. To prove this, note that we can expand f near p to get ± f(x , ,x )=f(p)+ x x H (x , ,x ) 1 ··· n i j ij 1 ··· n i,j X for smooth functions Hij. We can assume H is symmetric replacing Hij with (Hij +Hji)/2. The rest is a straightforward exercise in diagonalisation using the inverse function theorem. We can then use this to deduce the following important theorem. We will assume basic knowledge of Di↵erential Geometry.

6As we will see in the following section, this can alternatively be done by computing the homology groups using the Gysin sequence by the Poincar´econjecture for n 5.

11 Theorem (Reeb’s Theorem). Let M be a smooth compact manifold and f : M R a Morse function with exactly two critical points. Then M is homeomorphic to Sn.!

Proof: By compactness, f has a minimum x0 and a maximum x1 and so these are the only critical points and, by linear rescaling, we can assume that f(x0)=0,f(x1)=1. The Morse lemma gives us a neighbourhood U around x0 and coordinates such that f(v , ,v )=v2 + + v2, 1 ··· n 1 ··· n 2 2 since x0 is a minimum point. Thus dv1 + + dvn is a Riemannian metric , on U, which can be extended to M by partitions··· of unity. We can now let hf· be· i r the vector field defined by X, f = X(f)forallf : M R and vector fields X, h r i ! noting that this is singular precicely at x0 and x1. 2 Conisder the vector field f/ f defined on M x0,x1 and let t : M M be the associated flow, ar 1-parameterkr k family of di↵\{eomorphisms.} For any q !M, note that 2 df (t(q)) dt(q) f = , f = r 2 , f =1 dt dt r * f r + ⌧ kr k and so f(t(q)) = f(q)+t. Hence for each 0 " < 1 t,theflowmap" 1 1  n 1 induces a di↵eomorphism f [0,t] f [0,t+ "]. Now D = f [0, "] is a disc for suciently small ", and we define! : Dn M, q (q) ! 7! f(q)/" n n which retricts to a di↵eomorphism D˚ M x1 . Since @D x1 , this then descends to a homeomorphism Sn M!n. \{ } 7! { } ! To apply this to the Mk recall that it was defined via the clutching construction to be

3 3 Mk =(H S ) (H0 S )/ ⇥ t ⇥ ⇠ 4 4 where H = S N and H0 = S S are each identified with the quaternions for ease \{ } \{ } of notation. Note that, to identify H and H0, we have to “flip” the algebraic structure in 1 that u H⇥ corresponds to u (H0)⇥. More generally 2 2 i j i i 1 u vu 1 u (vu)u (u, v) (u0,v0)= u , i+j = u , ⇠ u ! u ! k k k k 3 since i + j =1.Defineafunctionf : H0 S R by ⇥ ! Re(v0) f(u0,v0)= (1 + u 2)1/2 k 0k 3 where Re(v0)denotestherealpartofthequaternionv0 S H0. Since the real part is the kernel of the action by SO(3), it is fixed by conjugation2 and✓ so

i i Re(u (vu)u ) Re(vu) f(u0,v0)= = (1 + u 2)1/2 (1 + u 2)1/2 k k k k which we take as our definition for f(u, v)for(u, v) H S3, thus extending f to a smooth 2 ⇥ map f : Mk R. We can then check by writing f out explicitly using coordinates that f ! has two critical points (u0,v0)=(0, 1) in the first chart and none in the second. Hence 7 ± Mk is homeomorphic to S by Reeb’s theorem.

12 3.2 Milnor’s Invariant Here we use Pontryagin numbers to construct an invariant for manifolds homeomorphic to S7 which is able to distinguish smooth structure. From this point onwards, we will take n-manifold to mean a connected compact orientable smooth n-dimensional manifold- with-boundary and closed to in addition mean that a manifold has no boundary. Let M be an closed 7-manifold with the homology of S7 and which bounds7 an 8-manifold B. Note that, since H3(M)=H4(M)=0,theinclusionmapi : M , B gives rise to an isomorphism i : H4(B,M) H4(B)bythelongexactsequencefor(!B,M). ! Note that, similarly to the case of closed manifolds, an orientation on B is determined by its fundamental class [B] H8(B,M). The relative cap product then gives a symmetric 2 bilinear form , : H4(B,M) H4(B,M)/(torsion) Z given by h · · i ⇥ ! x, y =(x ^ y)[B]. h i Since this form is symmetric, it has a well-defined signature which we denote (B). Ob- serve that there is a formula for the signature in the case of closed manifolds. The proof uses cobordism theory and will be postponed until the following chapter. Theorem (Hirzebruch’s Signature Theorem). If M is a closed 4n-manifold as above, then there is a polynomial Ln(x1, ,xn) Q[x1, ,xn] such that ··· 2 ··· (M)=L (p , ,p )[M], n 1 ··· n 4i 4n where pi = pi(TM) H (M)andeachmonomialinLn(p1, ,pn)liesinH (M). In particular L = 1 p and2 L = 1 (7p p2). ··· 1 3 1 2 45 2 1 Whilst B is not a closed manifold, we might wonder if expression was still true in a certain 4 8 sense. Consider the Pontryagin classes p1 = p1(TB) H (B)andp2 = p2(TB) H (B). 1 42 2 Now p can be pulled back along i to get i p H (B,M)andhenceaninvariant 1 1 2 1 2 (i p1) [B] Z. 2 8 8 However, there is no analogous quantity for p2 since i : H (B,M) H (B)isn’tnec- 1 1 1 2 ! essarily an isomorphism. So since “(B) 45 (7(i p2) (i p1) )[B]” is not properly defined (let alone equal to (B)), the best we can hope for is that they are equal when 1 we kill the i p2 term by dropping to a quotient ring. In particular, multiplying through 1 2 by 45 and reducing mod 7 gives 3(B)+(i p1) [B]mod7. Definition (Milnor Invariant). Let M be a closed 7-manifold as above. Then define (M)=2q(B) (B)=2(q(B)+3(B)) mod 7 1 2 for any 8-manifold B with boundary M, where q(B)=(i p1) [B]. Example. Let M = S7 be the standard 7-sphere and choose B = D8. Since H4(B,M)= H4(B)=0,thequadraticformH4(B,M) H4(B,M) Z is zero and so (M)=0. ⌦ ! The diculty comes in showing that (M)isindependentofthechoiceofB. If so then being a di↵eomorphism invariant follows from noting that a di↵eomorphism f : M M 0 8 ! can be extended to a di↵eomorphism between B and a suitible B0 with boundary M 0. 7In fact, as will we see in the following section, every such closed 7-manifold bounds such an 8-manifold. 8This uses that oriented cobordism is an equivalence relation on the category of manifolds, which we prove in the following section.

13 To show that this is independent of the choice of B, i.e. that the defect to the signature theorem holding for B1 and B2 are the same, we form a closed 8-manifold C = B1 M B2 by gluing along M and give it orientation compatible with that of B and B (i.e.t the 1 2 opposite orientation to the one on B2). Since C is closed, the signature theorem gives 2 that (C)=0(notingthati =idandsoq(C)=p1[C]) and so it remains to show that ⇤ (C)=(B ) (B ). 1 2 This follows immediately from the following lemma. Lemma. (i) (C)=(B ) (B ), and (ii) q(C)=q(B ) q(B ). 1 2 1 2 Proof: We start by considering all the relevant cohomology groups and the maps between them. We get the following diagram of isomorphisms:

4 h 4 4 H (C, M) H (B1,M) H (B2,M) ⇠ ⇠ ⇠ j i1 i2 4 k 4 4 H (C) H (B1) H (B2) ⇠ where h and k come from the Mayer-Vietoris sequence for relative cohomology and i1, i2 and j are the inclusions of M into B1, B2 and C respectively. This is commutative by naturality of the Mayer-Vietoris sequence. 4 1 4 (i) Let ↵i H (Bi,M)andthecorresponding↵ = jh (↵1 ↵2) H (C). 2 1 2 Then, since [C]=(jh )([B ] [ B ]) by construction, we get: 1 2 2 1 2 2 2 2 2 2 ↵ [C]=jh (↵ ↵ )[C]=(↵ ↵ )([B ] ( [B ])) = ↵ [B ] ↵ [B ]. 1 2 1 2 1 2 1 1 2 2 Hence the quadratic form for C splits as a direct sum of the quadratic forms for B1 and B2, i.e. the matrix for C is block diagonal consisting of B1 and B . The result then follows. 2 (ii) Note that inclusion f : B , C is an embedding and so (Df ,f) : TB TC i i ! i i i ! is a bundle map. By naturality, this gives that (fi)⇤p1(TC)=p1(TBi)for i =1, 2. By definition of k, this says that k(p (TC)) = p (TB ) p (TB ). 1 1 1 1 2 By commutativity, this is equivalent to 1 1 1 p (TC)=jh (i p (TB ) i p (TB )). 1 1 1 1 2 1 2 The formula dervied above then gives that what we want: 2 1 2 1 2 p (TC) [C]=(i p (TB )) [B ] (i p (TB )) [B ]. 1 1 1 1 1 2 1 2 2 This completes the proof that is a well-defined di↵eomorphism invariant. Hence by our previous example if M is di↵eomorphic to S7 then (M) = 0. The rest of this subsection will be devoted to computing (Mk). We start by considering a special case. 1 Example. To compute E01 we note that it is di↵eomorphic to H, the total space of the 1 tautological bundle over HP . This can be shown by writing down the obvious maps between the pairs of charts used to define each manifold and checking they agree. 1 1 2 1 4 1 We showed previously that p1(H)= 2e and p2(H)=e where e = e(H) H (HP ). 1 2 To write this is a more universal form, fix an isomorphism H4(HP ) H4(S4)andlet ⌫ H4(S4)betheimageofe under this map. Hence p (E )= 2⌫ and! p (E )=⌫2. 2 1 01 2 01 14 n n k Recall the Grassmannian Grk(R )= k-dimensional linear subspaces of R and its tautological bundle. Then it can be shown{ that every n-dimensional vector} bundle over Sk is a pullback of k. In particular, this construction gives a group homomorphism

4 FibS3 (S ) ij Eij Eij (u (R(Eij ))u)/ 7! 7! 7!

8 ⇡3(SO(4)) ⇡4(G4(R ))

4 4 where R , R(Eij) S is the vector bundle corresponding to the sphere bundle Eij. We use this! to establish7! the following. Proposition. p (E )=2(i j)⌫. 1 ij Proof: We start by showing that p1(Eij)islinearini and j, i.e. that (i, j) p1(Eij)is a group homomorphism. This is done by exhibiting the map as the composition:! 2 8 4 4 Z ⇡3(SO(4)) ⇡4(G4(R )) H (S ), ! ! ! 4 where the third map is [f] p1(f ⇤( )). To show this is a group homomorphism, 4 7! 4 note that p1(f ⇤( )) = f ⇤(p1( )) by naturality and so 4 4 4 (f g)⇤(p ( )) = f ⇤(p ( ))g⇤(p ( )) · 1 1 1 8 for f,g ⇡4(G4(R )). Hence p1(Eij)islinear. 2 Next note that if Eij is formed by taking quaternionic conjugate on each fibre, then p1(Eij)=p1(Eij) by earlier results. The transitions function between the two i j charts is (u, v) (u 1, u vu ) and conjugating both fibres gives u i+j 7! k k j i j i u¯ v¯u¯ u vu¯ (u, v¯) i+j = j i , 7! u u k k k k j i u vu 9 so Eij has transition function (u, v) j i , which shows that Eij = E j, i. u ⇠ 7! k k Hence p1(Eij)=p1(E j, i). By linearity, we can now write p1(Eij)=a(i j)⌫ for 4 4 some a Z and generator ⌫ H (S ). Finally a =2sincep1(E01)= 2⌫. 2 2 Remark. We can also show that e(Eij)=(i+j)⌫ using e(Eij)=e(E i, j)ande(E01)=⌫. Note that M has a corresponding disc bundle D4 , B S4 where the total space B k ! k ! k has boundary Mk, and so Mk is indeed of the form we have considered in this chapter. So far we know that p1(Mk)=2k⌫ and we want to compute p1(TBk). This is done using the following trick: Lemma. If ⇡ : E X is a smooth fibre bundle over a closed manifold X equipped with a Riemannian metric! on each fibre, then

TE = ⇡⇤(TX) ⇡⇤(E). ⇠

This can be proven by noting that the surjection D⇡ : TE ⇡⇤(TX)inducesashort exact sequence !

D⇡ 0 TvE TE ⇡⇤(TX) 0.

It can then be checked that this splits and that TvE = ⇡⇤(E). 9This is why we ignored the cases where i + j = 1 even though E is homeomorphic to S7. ij 15 Theorem. (M ) k2 1mod7. k ⌘ 4 Proof: Since Bk is a vector bundle, the zero section gives a homotopy equivalence Bk S . 4 ' Hence H (Bk) ⇠= Z and we can pick our orientation so that (Bk)=1. The embedding i : M , B induces a bundle map (Di, i) : TM TB and so k ! k k ! k i⇤(p1(TBk)) = p1(TMk). By the lemma above,

4 TM = ⇡⇤(TS ) ⇡⇤(M ) k ⇠ k 4 4 where ⇡ : Mk S is projection. Since H⇤(S ) = Z has no 2-torsion, this gives ! ⇠ 4 p (TM )=⇡⇤(p (TS )) ⇡⇤(p (M )) = ⇡⇤(0) ⇡⇤(2k⌫)=2k⇡⇤(⌫), 1 k 1 1 k 4 where we showed that p1(TS )=0inapreviousexample.Thiscanberewritten 1 4 as p1(TBk)=2k↵, where ↵ =(i⇤) (⇡⇤(⌫)) is a generator of H (Bk). Hence

1 2 2 2 (M )=2q(B ) (B )=2(i (2k↵)) [B ] 1=8k 1 k 1mod7 k k k k ⌘ 1 2 where (i ↵) [Bk] = 1 since ↵ corresponds to our choice of orientation. 2 7 Hence, if k 1mod7,thenMk cannot be di↵eomorphic to S . So we have shown that 7 6⌘ 7 S has at least 4 distinct di↵erentiable structures S , M0, M2 and M3, corresponding to the quadratic residues mod 7. This concludes our construction of exotic 7-spheres. 3.3 Higher Dimensions We will now more generally consider the case of closed (4k 1)-manifolds M homeomorphic 4k 1 to S . To generalise our invariant, let B be a 4k-manifold with boundary M. Note from before that inclusion M , B induces an isomorphism ! i : Hn(B,M) Hn(B) ! when Hn(M)=Hn+1(M)=0,i.e.whenever2 n 4k 3. Hence, if p , ,p denote   1 ··· k the Pontryagin classes of the tangent bundle of M, then p1, ,pk 1 can all be pulled n ··· back to give invariants in H (B,M). As before, we must eliminate the pk terms: Definition (Milnor Invariant). Let M be a closed (4k 1)-manifold as above. Then define 1 1 1 (M)= ((B) Lk(i p1, ,i pk 1, 0)[B]) Q/Z sk ··· 2 for any 4k-manifold B with boundary M, where s is the coecent of p in L (p , ,p ). k k k 1 ··· k Similarly to before, we could also clear denominators and reduce modulo the numerator of sk; we write this as . The proof that this is an invariant is the same as before. We will now try and generalise the above construction as much as possible to other values of k. In particular, wee want to find fibre bundles of the form

2k 2k 1 S , E S . ! ! 4k 1 where E is homeomorphic to S . As mentioned before, such fibre bundles only exist when 1 k 3, with structure groups SO(2), SO(4) and SO(8) respectively.   We will first consider the case k =1.SinceSO(2) is the group of rotations of R2,it 1 inherits the topology of S and so ⇡1(SO(2)) = Z. Thus the elements n Z precisely 2 16 1 2 correspond to fibre bundles S , En S . Alternatively, for n =0therotationgroup 1! ! 6 Gn = exp(2⇡i/n) : i Z S induces a fibre bundle { 2 }  S1/G = S1 , S3/G S3/S1 = S2. n ⇠ ! n ! ⇠ 3 3 Now S /Gn has to equal Em for some m, and it can be checked that in fact S /Gn = En 3 ⇠ (see [14]). These are examples of Lens spaces which have ⇡1(S /Gn) ⇠= Gn ⇠= Zn and so are not homeomorphic to S3 except when n = 1. It then follows that S1 , S3 S2 is the only smooth fibre bundle with total space homeomorphic± to S3. This agrees! with! the fact that there are no exotic 3-spheres. When k = 3, we can follow a similar process as in the k =2case.Indeedasimilarproof i j shows that ⇡7(SO(8)) = Z Z generated by u ij(u) : v u vu where multiplication ⇠ 7! 7! is octonionic. We thus get a set of 15-manifolds Eij. Since L = 1 (381p 71p p 19p2 +22p2p 3p4), the invariant is 4 14175 4 1 3 2 1 2 1 14175 71 19 22 3 (M)= (B) p p p2 + p2p p4 381 381 1 3 381 2 381 1 2 381 1 ✓ ◆ 1 where, by abuse of notation, we write pi to denote i pi and omit the [B] from the end of each term. For convenience, we multiply through by 381 and work mod 381 (rather than mod 1): (M)=78(B)+71p p +19p2 22p2p +3p4 mod 381. 1 3 2 1 2 1 We then similarly let Mk be Eij with i + j =1andi j = k. To compute p1, p2 and e 8 8 p3, let Bk be the corresponding disc bundle D , Bk S . By the long exact sequence ! !8 on homotopy groups, this gives that ⇡8(Bk)=⇡8(S ) = Z is the first non-vanishing 8 ⇠ homotopy group and so H (Bk) ⇠= Z is the first non-vanishing homology group by the Hurewicz theorem. Poincar´eDuality then gives that the only non-trivial homology groups i are H (Bk) ⇠= Z when i =0, 8, 16. This implies p1 = p3 =0,anditcanbecalculatedsimilarlytoabovethatp2 =6k↵ 8 8 where ↵ H (B,M)isagenerator.ThatH (B,M) ⇠= Z also gives that (B)=1with appropriate2 choice of orientation. Hence

1 2 2 (M)=78+19(i (6k↵)) [B ] = 78(1 k )mod381. k Since (78, 381) =e 3 and 381/3=127,wegetthat(M)=0i↵ k2 1mod127.Choosing ⌘ 15 suitable Morse functions also shows that Mk is homeomorphic to S . 2 e 7 Hence, if k 1mod127,thenMk is homeomorphic but not di↵eomorphic to S . This 127 1 6⌘ gives 2 =63distinctexotic15-spheres,eachcorrespondingtothequadraticresidues not equal to 1 (using that 127 is prime). We have thus shown that there exist 4distinctsmoothstructuresonS7 and 64 distinct smooth structures on S15. As we shall see in the final chapter, there are many more distinct smooth structures both on S7 and S15, as well as in other dimensions.

17 4 Cobordism

The goal of this section will be to give an introduction to cobordism, the study of manifolds up to the equivalence relation where two manifolds are equivalent if their disjoint union is the boundary of a manifold of one dimension higher10. Next we repay the debt created in the last chapter by proving Hirzebruch’s Signature Theorem and finally we state the h-cobordism theorem, showing how it can be used to deduce the Poincar´econjecture for n 5. Throughout this section, M will denote the oriented manifold M with opposite orientation and + will denote the disjoint union. Definition. An (oriented) cobordism between closed oriented n-manifolds M and N is a compact oriented manifold X such that

@X = M +( N) ⇠ is an orientation-preserving di↵eomorphism, where @X has the induced orientation. We say M and N are (oriented) cobordant or belong to the same (oriented) cobordism class. Note that we need compactness since otherwise M = @(M [0, )) and so all cobordism classes would be trivial. Following the conventions from earlier,⇥ 1 we now drop the words oriented and compact. We will need the following result from Di↵erential Geometry: Theorem (Collar Neighbourhood Theorem). If X is a smooth manifold, then there is an open set @X U X such that U = @X [0, 1) is a di↵eomorphism. ✓ ✓ ⇠ ⇥ This can be proven similarly to Reeb’s theorem by picking a Riemannian metric (using partitions of unity) and then considering the flow from an appropriate vector field. Lemma. Oriented cobordism is an equivalence relation on the category of manifolds.

Proof: For reflexivity, suppose f : M M 0 is a di↵eomorphism. Then X =[0, 1] ! ⇥ M [0, 1] M 0 has boundary M +( M 0), as required. Symmetry is obvious. t(id,f) ⇥ For transitivity, suppose M N and N R, so that M +( N) = @X and ⇠ ⇠ ⇠ 1 N +( R) ⇠= @X2 for (n+1)-manifolds X1 and X2. By applying the theorem above and restricting to components of the boundary, we can find open sets @X U with i ✓ i di↵eomorphisms fi : Ui N [0, 1). We can check that X3 = X1 f1(U1) f2(U2) X2 ! ⇥ [ ⇠ is a smooth compact manifold with M +( R) = @X , and so M R. ⇠ 3 ⇠ Let ⌦n denote the oriented cobordism classes of n- manifolds. Note that ⌦n forms an abelian group under + with 0⌦n =[M] for any closed manifold M. We also have a bilinear product ⌦m ⌦n ⌦m n, ([M], [N]) [M N] ⇥ ! ⇥ 7! ⇥ as M N is an (n + m)-manifold. Since (M N) R = (M R) (N R), this gives ⇥ t ⇥ ⇠ ⇥ t ⇥ ⌦ the structure of a graded ring with 2-sided identity ⌦0. In fact, ⌦ is graded ⇤ ⇤2 ⇤ commutative since M m N n = ( 1)mnN n M m is orientation-preserving. ⇥ ⇠ ⇥ Example. (i) ⌦0 ⇠= Z. This is since the boundaries of 1-manifolds are either or P +( Q), and so the sum of the signs of points is a complete cobordism invariant.; 1 2 (ii) ⌦1, ⌦2 =0sinceS = @D and ⌃g can be filled in by a connected sum of solid tori.

10With the operation of passing to the boundary, this is a generalised homology theory in that it satisfies all but one of the axioms of homology.

18 To compute these groups, we need to develop a cobordism invariant. Recall that for a 4n-manifoldM and a partition I =(i , ,i )ofn,theIth Pontryagin number is 1 ··· r

pI [M]=pi1 ir [M]=(pi1 (TM) pir (TM))[M]. ··· ··· Note that changing the orientation of M does not change the Pontryagin classes, but it does change the sign of the fundamental class and so changes the sign of pI [M]. Hence if pI [M] =0,thentheredoesn’texistanorientation-reversingdi↵eomorphism. In fact, it can be6 shown that, if M is the boundary of a (4n +1)-dimensionalmanifold,then pI [M] = 0 for all partitions I of n. We can now to show that pI is a cobordism invariant:

Proposition. For any partition I of n, we have a well-defined homomorphism ⌦4n Z given by [M] p [M]. ! 7! I Proof: It can be easily checked that pI [M + N]=pI [M]+pI [N]. To show well-defined, suppose [M]=[N]. Then there is a compact oriented (4n +1)-manifoldX such that M +( N) = @X. The above identity and lemma give that ⇠ p [M]+p [ N]=p [M +( N)] = p [@X]=0 I I I I for all partitions I. Noting p [ N]= p [N] then gives that p [M]=p [N]. I I I I This now gives us a way of proving that ⌦ =0forvariousn. n 6 Example. We can show ⌦4n =0byexhibitingaclosed4n-manifold with a non-vanishing Pontryagin number. In particular,6 as computed previously, we have that

2n 2n +1 2n +1 pI [CP ]= =0. i ··· i 6 ✓ 1 ◆ ✓ r ◆

2i1 2ir So for each partition I =(i1, ,ir)ofn, the products CP CP are non-zero ··· ⇥ ···⇥ in ⌦4n. This gives p(n)elementsin⌦4n and p(n) invariants to distinguish them, where p(n)isthenumberofpartitionsofn. By ordering the set of partitions appropriately, the corresponding matrix can be show to be upper triangular with non-zero entries on 2i 2i the diagonal and thus non-singular. Hence the CP 1 CP r represent linearly independent elements in ⌦ , and so rank(⌦ ) p(n). ⇥ ··· ⇥ 4n 4n Remarkably, Thom showed the following using the Thom space construction (see [15]):

2i1 2ir Theorem (Thom). The products CP CP , where i1, ,ir ranges over the ⇥ ···⇥ ··· partitions of n, are a basis for ⌦4n. In particular, rank(⌦4n)=p(n). 2 Example. (i) The torsion-free part of ⌦4 ⇠= Z is generated by CP . 2 2 4 (ii) The torsion-free part of ⌦8 = Z Z is generated by CP CP and CP . ⇠ ⇥ Furthermore, Terence Wall showed in 1960 that ⌦n is completely determined by the Pontryagin numbers and the Stiefel-Whitney numbers, which come from corresponding characteristic classes with Z2 coecients. This in turn shows that the ⌦n were the direct sum of copies of Z2 and, if n is a multiple of 4, copies of Z:

n 01234 5 67 8 9 1011 ⌦n Z 000ZZ2 00Z ZZ2 Z2 Z2 Z2

19 4.1 Hirzebruch’s Signature Theorem

Here we will use rank(⌦4n)=p(n) to deduce Hirzebruch’s Signature Theorem. We start by introducing the notion of a multiplicative sequence. 0 1 2 Let R be a commutative ring with unity and let A⇤ =(A ,A ,A , )beacommutative ⇡ ··· i graded R-algebra with unity. Also let A be the formal sums a0 + a1 + with ai A ⇡ ··· 2 and let A1 be the subgroup consisting of those sums with a0 =1.Forexample,wecan take R = Q and An = H4n(B; Q). Now consider a sequence of polynomials over R: K (x ),K (x ,x ),K (x ,x ,x ), 1 1 2 1 2 3 1 2 3 ··· where we let deg(xi)=i, and with the property that each Kn(x1, ,xn)ishomogeneous of degree n. Given a =1+a + a + A⇡, define K(a) A⇡ ···by: 1 2 ···2 1 2 1 K(a)=1+K (a )+K (a ,a )+ 1 1 2 1 2 ···

Definition. The Kn form a multiplicative sequence of polynomials if K(ab)=K(a)K(b) ⇡ for all R-algebras A⇤ and for all a, b A . 2 1 Example. Some examples of multiplicative sequences for general rings R are as follows. (i) Given ⇤,letK (x , ,x )=nx . This is a multiplicative sequence with 2 n 1 ··· n n K(1 + a + a + )=1+a + 2a + 1 2 ··· 1 2 ··· i i.e. K simply replaces ai with ai. 1 (ii) Insisting that K(a)=a defines a multiplicative sequence: x ,x2 x , x3 +2x x x , 1 1 2 1 1 2 3 ··· which, in general, takes the form

(i1 + in)! i1 in Kn(x1, ,xn)= ··· ( x1) ( xn) . ··· i1! in! ··· i1+2i2+ +nin=n X··· ···

Now consider any t A⇤ of degree 1. Then we have 2 K(1 + t)=1+K (t)+K (t, 0) + =1+ t + t2 + 1 2 ··· 1 2 ··· n where n is the coecient of x1 in Kn(x1, ,xn). The following lemma says that this relation is enough to determine the entire multiplicative··· sequence..

2 Lemma (Hirzebruch). Given a formal power series f(t)=1+1t+2t + with i R, there is a unique multiplicative sequence K with coecients in ⇤ such··· that 2 { n} K(1 + t)=f(t).

We say K belongs to f(t). { n} Proof: For uniqueness, let A⇤ = ⇤[t , ,t ] where deg(t )=1andlet 1 ··· n i =(1+t ) (1 + t ) A⇡. 1 ··· n 2 1 Since K is multiplicative, we have

K()=K(1 + t ) K(1 + t )=f(t ) f(t ). 1 ··· n 1 ··· n 20 Now =1+ + , where are the elementary symmetric polynomials, 1 ··· n i so Kn(1, , n)isdeterminedbyf. Hence f determines Kn since it is a basic result from··· Galois Theory that , , are algebraically independent. 1 ··· n For existence, let I = i , ,i be a partition of n and let 1 ··· r s ( , , )= ti1 tir , I 1 ··· n 1 ··· r where the sum over all the distinct monomialsX that are produced by permuting the tis. It can be shown that the sI form an additive basis for the set of symmetric polynomials. Now let

K ( , , )= s ( , , ), n 1 ··· n I I 1 ··· n I X which gives a well-defined sequence since the i are algebraically independent, as above. To show the Kn are multiplicative, we can easily check that

sI (ab)= sH (a)sJ (b), HJ=I X where HJ denotes adjoining of partitions. Hence

K(ab)= I sI (ab)= sH (a)sJ (b)= H sH (a)J sJ (b)=K(a)K(b). I I HJ=I H,J X X X X Example. Applying the above lemma to our examples from before, we get: (i) (x , x , x , ) 1+t, since K(1 + a )=1+a . 1 2 3 ··· $ 1 1 2 2 1 (ii) ( x ,x x , ) 1 t + t , since K(1 + a )=(1+a ) . 1 1 2 ··· $ ··· 1 1 Consider now a multiplicative sequence K with rational coecients and a closed 4n- manifold M. Definition. The K-genus of a 4n-manifold M is given by: K[M]=K (p , ,p )[M], n 1 ··· n 4i where pi = pi(TM) H (M; Q)isthePontryaginclassand[M] is the fundamental class. If 4 doesn’t divide the2 dimension of M, we define K[M] = 0. In turns out that the K-genus is a cobordism invariant which respects the disjoint union and cartesian product of manifolds. This is summarised in the following lemma. Lemma. If K is a multiplicative sequence with rational coeents, then [M] K[M] n 7! defines a ring homomorphism ⌦ Q and hence an algebra homomorphism ⌦ Q Q. ⇤ ! ⇤ ⌦ ! Proof: We previous had that pI [M + N]=pI [M]+pI [N]andpI [@X] = 0 and so we need only show that K[M N]=K[M]K[N]. Indeed note that ⇥ p(T (M N)) = p(TM TN) p(TM)p(TN)modulo2-torsion, ⇥ ⌘ so are equal with Q coecients. Since K is multiplicative, we have K(p(T (M N))) = K(p(TM))K(p(TN)). ⇥ Comparing top dimenional terms and noting that [M N]=[M][N] then gives that K[M N]=K[M]K[N], as required. ⇥ ⇥ 21 We note that the signature of a manifold satisfies the same properties. Lemma. The map [M] (M) defines a ring homomorphism : ⌦ Z and hence an 7! ⇤ ! algebra homomorphism ⌦ Q Q. ⇤ ⌦ ! Proving this amounts the showing the following for manifolds M and N. 1. (M + N)=(M)+(N). 2. (M N)=(M)(N). ⇥ 3. If M bounds, then (M)=0. The first part is immediate from noting that the matrix for M + N is block diagonal consisting of M and N. The second part can be proven using the K¨unneth isomorphism and the final part using Poincar´eduality. Whilst illuminating, we will omit the arguments for brevity and direct the reader to [17]. Our goal is now to find a multiplicative sequence L such that (M)=L[M] for any 4n-manifold M or equivalently to find the corresponding power series f(t). Once we find such an f, showing that it works is an exercise in finding the nth term in a Taylor Series11. Here however we take special care to show how such an f can be derived without any previous knowledge. First note that, since M (M)andanyM L[M] both give rise to algebra homo- 7! 7! morphisms ⌦ Q Q, we need only check equality on a set of generators for ⌦ Q, ⇤ ⌦ ! 2k ⇤ ⌦ i.e. on the complex projective spaces CP for 1 k n.   n 2 2n 2k Using a e(C) H (CP )asageneratorgivesthat(CP )=1sincethecorrespond- 2 2 2n ing matrix sends 1 a2n[CP ] = 1, by earlier calculation. So we require 7! 2k 2k Lk(p1, ,pk)[CP ]=L[CP ]=1 ··· 2k 2k 2 2k+1 for all k 1, where pi = pi(T CP ). Since p(T CP )=(1+a ) we know that if f(t)=L(1 + t), then

(f(a2))2k+1 = L(1+a2)2k+1 = L(1+p +p + )=1+L (p )+ +L (p , ,p )+ 1 2 ··· 1 1 ··· k 1 ··· k ··· 2k 2k and so Lk(p1, ,pk)=a where is the coecient of t in the Taylor expansion ··· 2k of (f(t2))2k+1. Hence the above gives that a2k[CP ] = 1 and so =1byearlier calculations. Hence we need a formal power series f(t)suchthat

(f(t2))2k+1 = a + a t2 + + t2k + 0 2 ··· ··· for all k 1. To find such a power series, we appeal to a classical formula which relates the Taylor series of a power of a function f to the Taylor series of the inverse of x/f(x): Theorem (Lagrange-B¨urmannformula). Let f be a formal power series with non-zero constant term and '(t)=t/f(t). If f(t)n = a + a t + , then o 1 ··· 1 an 1 n ' (t)=b + b t + + t + 0 1 ··· n ···

The proof is an exercise in computing residues and so is omited for brevity. Given our f above, the corresponding ' must be an odd function and consist of only odd terms.

11This can be achieved by computing residues and such a treatment can be found in [15].

22 2k+1 1 Furthermore, the coecient of t must be 2n+1 , so formal integration gives

1 1 3 1 5 2 4 1 ' (t)=t + t + t + = (1 + t + t + )dt = dt 3 5 ··· ··· 1 t2 Z Z 1 1 1 1 1+t = + dt = log and finally 2 1+t 1 t 2 1 t Z ✓ ◆ ✓ ◆ 1 1+'(t) 1+'(t) e2t 1 log = t = e2t '(t)= =tanh(t). 2 1 '(t) ) 1 '(t) ) e2t +1 ✓ ◆ Hence, by the formula above, we get f(t2)=t/ tanh(t)andsof(t)=pt/ tanh pt. Hence, by using general formulae to find the nth term of this series, we have shown:

Theorem (Hirzebruch’s Signature theorem). If Ln be the multiplicative sequence of polynomials belonging to the power series { } pt 1 1 2 22nB =1+ t t2 + t3 + 2n tn + tanh pt 3 45 945 ··· (2n)! ··· where the Bn are the Bernoulli numbers, then the signature (M)ofany4n-manifold M is equal to the L-genus L[M]. This in particular shows that the L-genus is an integer-valued homotopy invariant. Example. Note that each term of the sequence gives a constraint on the Pontryagin numbers pi = pi(TM). To compute these polynomials, we will use the formula L ( , , )= s ( , , ) n 1 ··· n I I I 1 ··· n from the proof that multiplicative sequencesP correspond to formal power series, where =1/3, = 1/45, =2/945, are the coecients in the Taylor series above. 1 2 3 ··· 1 (i) That L1 = 1p1 = 3 p1 follows immediately.

(ii) The partitions of 2 are (1, 1), (2), and 1 = p1 + p2, 2 = p1p2, so 1 1 1 L ( , )=2p p + (p2 + p2)= (2 2 )= (7 2) 2 1 2 1 1 2 2 1 2 9 2 45 1 2 45 2 1 Hence L = 1 (7p p2). 2 45 2 1 (iii) We have 1 = p1 + p2 + p3, 2 = p1p2 + p2p3 + p3p1 and 3 = p1p2p3, so

3 2 2 3 3 3 L3(1, 2, 3)=1p1p2p3 + 12 (pipi+1 + pi pi+1)+3(p1 + p2 + p3) i 1 1 X 2 = ( 3 )+ (3 3 +3 ). 27 3 135 1 2 3 945 1 1 2 3 Hence L = 1 (62p 13p p +2p3). 3 945 3 1 2 1 (iv) Similarly we can compute that L = 1 (381p 71p p 19p2 +22p2p 3p4). 4 14175 4 1 3 2 1 2 1 As above, computing Ln amounts to finding expressions for the each sI in terms of the n 2n i. The coecient of p1 in each Ln is n =2 B2n/(2n)! by construction, and using such formulae we can also get that the coecient of pn is 2n 2n 1 2 (2 1)B /(2n)! 2n However, surprisingly little is own about the other coecients. For example, it is not even known whether or not every monomial has non-zero coecient.

23 4.2 h-Cobordism Theorem We end this section by outlining an important result in high-dimensional topology, namely the h-cobordism theorem. This will be a key ingredient in the classification of exotic spheres, though for now our aim will be to show that this implies the Poincar´econjecture for n 5. We end by showing how this will allow us to develop tools for proving manifolds are homeomorphic to spheres, which we use in the following sections. We start by refining the notion of cobordism that we considered previously. Definition. An (oriented) cobordism X between closed n-manifolds M and N is a h- cobordism if the inclusion maps M , X and N , X are homotopy equivalences. We say a h-cobordism is simply-connected! if M, !N and X are all simply-connected. It is easy to see that h-cobordism between closed manifolds and simply-connected h- cobordism between simply-connected closed manifolds are both equivalence relations. In 1962, Stephen Smale proved12 the following. Theorem (h-Cobordism Theorem). Let M and N be simply-connected closed n-manifolds and X asimply-connectedh-cobordism between them. If n 5, then X is di↵eomorphic to M [0, 1]. In particular, M and N are di↵eomorphic. ⇥ This is done by finding a Morse function on X and showing that this decomposes X in to a number of “handles” of di↵erent types. This handle decomposition can then be put into a canonical form where we can then attempt to “cancel” handles. A careful analysis using the Whitney trick then reveals that having suciently high dimensions is necessary to ensure that the resulting manifold remains simply-connected. Hence to show two simply connected n-manifolds M and N for n 5aredi↵eomorphic, it suces to show that they are (simply-connectedly) h-cobordant. Indeed, this led to many of the key questions in simply-connected topology being solved over the coming years13. For convenience (but not necessity), we record the following results which allow us to avoid working with maps M,N , X ! Theorem (Poincar´eDuality for Cobordisms). If X is a cobordism between n-manifolds n+1 i M and N, then Hi(X, M) ⇠= H (X, N). This can be proven using handles, as in the proof on the theorem above. Lemma. If X is a simply-connected cobordism between simply-connected manifolds M and N, then X is a h-cobordism i↵ H (X, M)=0. ⇤ Proof: By the relative version of the Hurewicz theorem, we have that ⇡ (X, M)=0.By ⇤ noting that the relative homotopy groups fit into a long exact sequence (as do the homology groups), this gives that the inclusion M , X induces isomorphisms ⇡ (M) ⇡ (X). The Whitehead theorem tells us! that a map which induces ⇤ ⇤ isomorphisms! on all homotopy groups must be a homotopy equivalence, and so M , X is a homotopy equivalence. ! By the theorem above, we have that H⇤(X, N)=0andsoH (X, N)=0also. ⇤ Hence N , X is also a homotopy equivalence. The converse follows easily by reversing the! above argument.

12Actually he originally proved this for n 7, though the result was later expanded. For this, he was awarded the Fields Medal 1966. 13For example, Terence Wall completed a classification of (n 1)-connected (2n 1)-manifolds.

24 We can use this to give the above application. Theorem. Let M be a simply-connected n-manifold with simply-connected boundary, for n 5. Then M is di↵eomorphic to Dn i↵ M has the homology of a point. Proof: Suppose M has the homology of a point and let D0 M @M be an n-disc ✓ \ n 1 embedded in a chart of M. Then M D˚ is still simply-connected since @D = S \ 0 0 ⇠ and n>2, and so is a simply-connected cobordism between @M and @D0.To show this is a h-cobordism, we need that H (M D˚0, @D0)=0bytheabove. ⇤ \ Since M and D0 both have the homology of a point, it follows that H (M,D0)=0. ⇤ To apply excision we have @D D0 D˚ for a larger disc D D0 . Hence 0 ' 0 \ 0 0 ✓ 0 ˚ ˚ ˚ H (M D0, @D0)=H (M D0,D0 D0)=H (M,D0)=0. ⇤ \ ⇤ \ \ ⇤

By the h-cobordism theorem, M D˚0 is di↵eomorphic to @D0 [0, 1]. There are many ways to show this can be extended\ to a di↵eomorphisc between⇥ M and Dn. For example, note that M and Dn are both the composition of the cobordism from to @D0 (given by D0)andthecobordismfrom@D0 to @M (given by M D˚0). Since; there is a unique smooth structure on this composition of two cobordisms,\ we must have that M is di↵eomorphic to D. Hence we have shown that any n-manifold homeomorphic to Dn must be di↵eomorphic to Dn for any n 5. This is the statement that there are no exotic n-discs for n 5. Even better, this is then powerful enough to resolve all but a handful of cases of one of the longest standing theorems in topology. Theorem (Poincar´eConjecture). Let M be a closed simply-connected n-manifold with the homology of Sn. If n 5, then M is homeomorphic to Sn. Proof: Let D M be an n-disc embedded in a chart of M. Now, by Poincar´eDuality (for manifolds-with-boundary)✓ as well as using the excision argument from the proof above, we have

n i n i H (M D˚) = H (M D,˚ @D) = H (M,D). i \ ⇠ \ ⇠ n i n i The long exact sequence on relative homology gives that H (M,D) ⇠= H (M) ⇠= n i n 0 ˚ H (S )wheni = n,andH (M,D) ⇠= 0. So M D has the homology of a point and, by the result6 above, M D˚ is di↵eomorphic\ to the n-disc. Hence M is a the union of two n-discs glued along\ a di↵eomorphism of their boundaries, i.e. M is a twisted sphere, which we previously showed is homeomorphic to Sn. This is often stated in the following slightly weaker form. Corollary. Let M be a closed n-manifold. If n 5, then M is homotopic to Sn i↵ M is homeomorphic to Sn. We will now give a few applications of this result. We start by giving an alternate proof 7 that the spheres Mk constructed in the previous section are indeed homeomorphic to S . 3 4 Proposition. If Eij is the sphere bundle S , Eij S constructed in the previous 7 ! ! section, then Eij is homeomorphic to S i↵ i+j = 1. In particular Mk is homeomorphic to S7. ± Proof: Firstly, the long exact sequence associated to the fibre bundle gives easily that ⇡1(Eij)=0.Secondly,theGysinsequenceforthecorresponding4-dimensional

25 vector bundle then gives

^e(E ) 3 0 4 · ij 4 4 4 0 H (Mk) H (S ) H (S ) H (Mk) 0

1 4 3 4 i i 7 since H (S )=H (S )=0,andalsogivesthatH (Mk)=H (S )foralli =0, 3, 4. 6 7 Since Eij is simply-connected, the result above gives it is homeomorphic to S i↵ 3 4 H (Mk)=H (Mk)=0i↵ the middle map is an isomorphism. We previously 4 4 calculated that e(Eij)=(i + j)⌫, where ⌫ H (S )isagenerator,andsothe middle map is an isomorphism i↵ i + j = 1.2 ± Remark. This actually follows from the Whitehead theorem which we used earlier in this section. In particular, Hurewicz implies ⇡7(Mk)=H7(Mk)=Z and the generator gives a map S7 M which we can show is an isomorphism on all homotopy groups. ! k We conclude this section by extracting the key ingredients of the above proof. In par- ticular, consider a closed (n 2)-connected (2n 1)-manifold M. Then the Hurewicz theorem implies that Hi(M)=0forall1 i n 2. Since all these homology groups are free, Poincar´eduality then tells us this holds for n +1 i 2n 2andalsothat   H2n 1(M)=H0(M)=Z. Hence we need only worry about the (n 1)th homology group in this case. Explicitly, we have: Corollary. If M is a closed (n 2)-connected (2n 1)-manifold, then M is homeomorphic 2n 1 to S i↵ Hn 1(M)=0. In the following two sections, we give two di↵erent constructions of (n 2)-connected (2n 1)-manifolds M which bound (n 1)-connected manifolds B and which we claim 2n 1 are exotic spheres. To show they are homeomorphic to S , consider the long exact sequence for the pair (B,M). Since M is (2n 1)-connected, this reduces to i 0 Hn(M) Hn(B) ⇤ Hn(B,M) Hn 1(M) 0, where we use Poincar´eduality and the universal coecients formula to deduce that n 1 Hn+1(B,M) = H (B)=0.Bytheconditionabove,weknowthatM is homeo- 2⇠n 1 morphic to S i↵ the middle map i is an isomorphism. We get the same situation for ⇤ cohomology by dualising to f ⇤.

Now in the example of Mk above, we used that the bounding manifold B was the total n n space of a fibre bundle ⇡ : B X to show that i⇤ : H (B,M) H (B)inducesamap ! ! Hn(B,M) i⇤ Hn(B) ⇠ ⇡⇤ ⇠

n d n H (X) H (X) where d is the dimension of the bundle and is the Thom isomorphism. We then identified the bottom map as cupping with the Euler class of the fibre bundle and thus were able to show that M was homeomorphic to asphere i↵ e(M)= 1. ± The examples we will consider in the following two sections will not be sphere bundles, but in both cases we alter i to give a linear map Zm Zm for some m and note that this is an isomorphism if the corresponding matrix M is! unimodular, i.e. det(M)= 1. ±

26 5 Plumbing

Here we give a brief overview of a di↵erent method of constructing exotic spheres, also introduced by Milnor14. We introduce an operation on disc bundles over manifolds known as plumbing, and show that this operation can be used to construct manifolds with a given intersection form. The basic construction is as follows. 5.1 Plumbing Disc Bundles For i =1, 2, consider disc bundles

n ⇡i Di Ei Ni, where the Ni are closed n-manifolds and the Ei are oriented compatibly with the Ni. Let xi Ni and pick an n-disc xi Di0 Ni that is contained is a trivialising open set of the 2 1 2 ✓ disc bundle. Then ⇡ (x ) D0 D under the trivialisation around D0, for i =1, 2. i i ✓ i ⇥ i i Now pick canonical di↵eomorphisms h : D10 D2 and k : D1 D20 such that h+,k+ ± ± and h ,k are orientation-preserving and orientation-reversing! respectively.! Definition (Plumbing). The result of plumbing together E and E with sign 1is 1 2 ± E 2 E = E E / , 1 2 1 t 2 ⇠ where (x, y) D10 D1 is identified with (k (y),h (x)) D20 D2, for some sign . 2 ⇥ ± ± 2 ⇥ ± More generally, if m 1, the result of plumbing together m points with sign 1isgiven ± j by repeating the above for m distinct pairs of points. In particular, pick points pi Di j j j j j 2 j for j =1,...,m, disjoint discs Di0 around each pi ,mapsh : D10 D2, k : D1 D20 j j j ±j ! ± ! and then identify (x, y) D10 D1 with (k (y),h (x)) D20 D2 for each j =1,...,m. 2 ⇥ ± ± 2 ⇥ 1 Example. We will consider the where case n =1,N1 = N2 = S and E1 = E2 = 1 1 S [0, 1]. Here the neighbourhoods D0 correponds to arcs of S and so the identification ⇥ i of the squares [0, 1] Di0 corresponds to the diagram below. The picture below is in general a good way to⇥ picture plumbing disc bundles in higher dimensions.

Figure 2: Two disc bundles E1 and E2 attached to form E1 2 E2.

14This construction goes back at least to [3], a set of particularly illuminating unpublished notes written by Milnor in 1959. A more recent exposition of this material can be found in [19].

27 This can be made into a smooth manifold by “straightening out the angles” using bump functions at the intersection points, and it can be given an orientation compatible with N and also either N if n is even or N if n is odd. 1 2 2 It can be shown that E1 2 E2 is an n-disc bundle over an appropriate n-manifold. Also observe the e↵ect of plumbing on the boundary.

Proposition. If @E1 and @E2 are (n 2)-connected, then @(E1 2 E2)istheunionoftwo (n 2)-connected sets whose intersection is (n 2)-connected. Proof: We start by noting that noting that

@(E 2 E )=(@E D0 @D ) (@E D0 @D ) 1 2 1 \ 1 ⇥ 1 [ 2 \ 2 ⇥ 2

which is clear from the picture above. Now @Ei Di0 @Di @Ei @Di and, since n 1 \ ⇥ ' \ S = @D , @E is a codimension n embedding, we know that ⇠ i ! i ⇡ (@E D0 @D ) ⇡ (@E )=0 j i \ i ⇥ i ! j i is an isomorphism for j n 2, and so (@E D0 @D )is(n 2)-connected.  i \ i ⇥ i To see that intersection is (n 2)-connected since, the picture above gives that n 1 n 1 (@E D0 @D ) (@E D0 @D )=@D0 @D = S S 1 \ 1 ⇥ 1 \ 2 \ 2 ⇥ 2 1 ⇥ 1 ⇠ ⇥ n 1 n 1 n 1 n 1 and we have that ⇡ (S S ) = ⇡ (S ) ⇡ (S )=0forj n 2. j ⇥ ⇠ j ⇥ j  Now note that we can consider Ni as lying in E1 2 E2 by mapping into Ei along the zero section followed by inclusion. If E and E are plumbed together at m points p , ,p , 1 2 1 ··· n then N1 and N2 have intersection N1 N2 = p1, ,pn . Since E1 2 E2 has dimension 2n, this is transverse with sgn(p )= \1foreach{ i, depending··· } on the sign of the plumbing. i ± Now let W = E1 2 E2 and consider the intersection product

1 1 Hn(W ) Hn(W ) H0(W ) = Z, (x, y) x y = DW (D (x) ^ D (y)) ⇥ ! ⇠ 7! · W W n where DW : H (W, @W ) Hn(W )isthePoincar´edualitymap.Tocomputetheinter- section product of N and!N , let i : N , W and i : N , W be as above and note 1 2 1 1 ! 2 2 ! that push forward the fundamental classes [N1]and[N2] givest elements in Hn(W ). The following standard result then gives this in terms of the signs of the intersection points. Lemma. (i1) [N1] (i2) [N2]= sgn(x). ⇤ · ⇤ x N1 N2 2X\ Since E1 2 E2 is still a disc bundle of the same form, we can carry out this process for k disc bundles Ei Ni for i =1, ,k. In particular, for any mij Z for i = j,we can build a 2n-manifold! W with submanifolds··· N , ,N each with intersection2 6 number 1 ··· m (i1) [N1] (i2) [N2]=nij, i.e. plumb together mij points with sign sgn mij. The case ⇤ ⇤ when i =·j is described the the following result.| | Lemma. (i1) [N] (i2) [N]=e( N W )[N]. ⇤ · ⇤ V ✓ This is proven by considering a tubular neighbourhood of N in W and using the Hopf index theorem. The proofs of these lemmas can be found in [19]. We can now use this to find manifolds which attain any intersection form possible.

28 Theorem. Let M be an k k symmetric matrix with integer entries and even entries on the diagonal. The for every⇥n 2, there is a 4n-manifold W with boundary such that (i) The matrix corresponding to the intersection form H2n(W ) H2n(W ) Z is M. ⇥ ! (ii) W is (2n 1)-connected, @W is (2n 2)-connected and H (W )isfreeabelian. 2n 2n 2n Proof: Let M =(mij), where mij = mji and mii =2i. Consider spheres S1 , ,Sk their tangent bundles ··· 2n 2n 2n R , TSi Si ! ! 2n 2n which have associated disc and sphere bundles D(TSi )andS(TSi )respectively. 2n Since S(TSi )canbebuildfromtheclutchingconstruction,itmustcorrespond 2n to an element in ⇡2n 1(SO(2n)) from our earlier work. Let iS(TSi ) denote the sphere bundle corresponding to i times this element in ⇡2n 1(SO(2n)), and let 2n Ei = iD(TSi ) be the corresponding disc bundle. Now note that, by the relation between the Euler class and Euler characteristic given earlier, we have

e(TS2n)[S2n]=(S2n)=1+( 1)2n =2 i i i 2n 2n 2n and this can be extended to show that e( S2n E )[Si ]=e(iTSi )[Si ]=2i. V i ✓ i Now let U = E1 2 2 Ek, where each Ei and Ej are plumbed together at mij points with sign sgn···m , when i = j. This gives the right matrix o↵ the diagonal| | ij 6 and on the diagonal gives 2i = mii by our calculation above. To show there are no 2n 2n other entries inthe matrix, i.e. that H2n(U)hasbasis(ij) [Sj ] where ij : Sj , U, ⇤ 2n ! it suces to note that U deformation retracts onto i Si / where attaches 2n 2n ⇠ ⇠ points on the spheres in such a way that S S = mij . i \ j S| | By our earlier remark about the e↵ect of Plumbing on the boundary of the mani- 2n fold, using that the Si are (2n 1)-connected, we have that @U is the union of k di↵erent(2n 2)-connected sets with a number of (2n 2)-connected intersections. Mayer-Vietoris then gives that Hi(X)=Hi(@X)=0forall2 i 2n 2for each connected component of U, where we note that n 2.   It can then be shown, by a series of attachments on the boundary, that U can be extended to a manifold W which is simply connected and has Hi(W )=Hi(U)for all i and Hi(@W )=Hi(@U)foralli 2n 2. The still has intersection matrix M and, by the Hurewicz theorem, W is (2n 1)-connected, has (2n 2)-connected boundary and has free 2n-th homology. Since @W is a (2n 2)-connected (4n 1)-manifold and bounds a (2n 1)-connected 4n-manifold W , we follow the approach outlined in the previous section to characterise 4n 1 when is homeomorphic to S . In particular, we have that the long exact sequence for the pair (W, @W )gives

i @ 0 H2n(@W ) H2n(W ) ⇤ H2n(W, @W ) H2n 1(@W ) 0, where i : @W , W is inclusion. Now note that Poincar´eduality gives an isomorphism 1 ! 2n DW : H2n(W, @W ) H (W )andtheuniversalcoecients formula gives an isomor- 2n ! phism H (W ) Hom(H2n(W ), Z). ! We can thus characterise whether or not this map is an isomorphism as follows.

29 4n 1 Theorem. If W is as above, then @W is homeomorphic to S i↵ det(M)= 1. ± Proof: Consider the map ↵ : H2n(W ) Hom(H2n(W ), Z) defined by the diagram below. ! i H2n(W ) ⇤ H2n(W, @W )

1 ⇠ DW ↵ H2n(W )

⇠ UCT

Hom(H2n(W ), Z)

1 2n Now ↵ sends x H2n(W )toDW (x) H (W )andthentothemapy 1 2 1 2 7! DW (DW (x) ^ DW (y)) in Hom(H2n(W ), Z), which can be checked by running through the proof of the universal coecients theorem and showing this definition works. So ↵ corresponds to the intersection product on H2n(W )andsoisrepre- 2n 2n 4n 1 sented by the matrix M : Z Z . Hence @W is homeomorphic to S i↵ i ⇤ an isomorphism i↵ M is unimodular,! i.e. det(M)= 1. ± 5.2 Examples of Exotic Spheres Our aim now is to find unimodular matrices M of the above form. We would then know 4n 1 that the corresponding smooth (4n 1)-manifold @W would be homotopic to S . Since this comes with a bounding manifold W , we would then have some hope of showing that @W is an exotic sphere using the characteristic class invariants established earlier. Consider the following 8 8matrixM (left) which is of the required form. By successive row and column operations,⇥ we get the diagonal matrix D (right).

21 20 2 3 3 121 0 2 0 2 0 1 0 4 1 0 4 1 121 0 3 0 3 B C B 5 C B 5 C B 121 C B 0 4 0 C B 4 C B C B 6 C B 7 C B 12101C ! B 0 5 101C ! B 10 C B C B C B 4 C B 1210C B 1210C B 7 C B C B C B 1 C B 0120C B 0120C B 4 C B C B C B C B 1002C B 1002C B 2C B C B C B C @ A @ A @ A This shows that M has signature 8 and determinant 1. Hence, for any n>1, we get a 4n 1 corresponding 4n-manifold W with boundary @W homeomorphic to S . In the notation of the section on higher dimensions at the end of the third chapter, we now want to compute

1 1 1 (@W )= ((W ) Ln(i p1, ,i pn 1, 0)[W ]) Q/Z, sn ··· 2 where sn is the coecent of pn in Ln(p1, ,pn). The important thing here is that, since ···i W is (2n 1)-connected, we have that H (W, @W ) = H4n i(W )=0fori =0, 2n, 4n. In ⇠ particular 6 1 4i i p H (W, @W )=0 i 2 when 4i =0, 2n, 4n and so all Pontryagin classes in the expression above, except possibly 1 6 1 i pn/2 in the case where n is even, are zero. In fact i pn/2 =0bynotingthatthe

30 2n Pontyagin classes of TSi are all zero, by previous calculations, and we can check that the result of Plumbing then restricting to the boundary doesn’t change this fact. Combining with the fact the signature is 8 from the matrix above, we then have that

(@W )=8/sn Q/Z. 2 Finally, this can be computed by the formula

2n 2n 1 s =2 (2 1)B /(2n)! n 2n 4n 1 which we stating in this previous section. The denominator of (@W )isthen7,31, 127 and 73 when n =2,3,4and5respectively.Itturnsoutthatthedenominatoris always > 1andso@W is always an exotic sphere for any n 2. We call this the Milnor sphere and denote it by ⌃M . In particular, this shows that there are exotic spheres of 4n 1 arbitarirly large dimension of the form S . An interesting observation can be made by representing the construction of W by a graph, 2n 2n namely by letting the vertices be the spheres S1 , ,Sk and drawing m appropriately 2n 2n ··· | | signed edges between Si and Sj whenever they are plumbed together at m points. In this case all attachments are made by single points with sign 1 and give the| graph|

•

• • • • • • • which can be identified as the dynkin diagram of the Lie group E8. These links can be explained by considering the theory of symmetric matrices with integer entries and even entries on the diagonal. These constructions can be taken much further and can in fact be used to construct exotic 4n+1 spheres of the form S also. In particular the Kervaire sphere, which we write as ⌃K , can be constructed by plumbing together two copies of D(TS2n+1)insuchawaythatthe associated intersection form is 01 10 ! which has determinant 1. This gives a (4n +2)-manifoldK and @K is indeed an exotic sphere for some values of n. In fact, determining which n have @K4n+2 an exotic sphere has turned out to be closely related to many other problems in algebraic topology. This is known as the Kervaire invariant problem and very recent breakthroughs have shown that it is true precisely when n =2,6,14,30,62andpossibly162(theonlyunknown case). This gives us exotic spheres in dimensions 9, 25, 57, 121, 249 and possibly 649. An interesting application of this comes from resticting our attention to the 9-dimensional Kervaire sphere ⌃K . Let W be the 10-manifold bounding ⌃K in the above construction. If f : ⌃ S9 is a homeomorphism, then let K ! W = W D10 [f be the topological 10-manifold form by gluing W and D10 along f. Amazingly, the fact that ⌃K is an exotic sphere can be used to show that this topological manifold does not admit any smooth structure. For a proof, the reader is directed to [5]. We will return to both the Milnor spheres and Kervaire spheres in the following chapters.

31 6 Brieskorn Varieties

Here we discuss a construction of Exotic Spheres which can be build out of taking a complex hypersurface in Cn+1 and intersecting it with a very small (2n+1)-sphere around aparticularpointinV . Explicitly, let f(z , ,z )beapolynomialincomplexvariablesandlet 1 ··· n+1 1 V = V (f)=f (0)

n+1 be the zero set of f, equipped with the subspace topology from C . Pick z0 V , " > 0 2 and define the link at z0 to be K = V S , \ " n+1 where S" = z C : z z0 = " is the small sphere of radius " around z0. { 2 k k }

Figure 3: An illustration of the link around a singularity K.

From now on, we will assume deg f>1. This ensures we can find i for which @f/@zi is non-constant. Hence any critical point of f is a root of @f/@zi which shows that the set ⌃(V )ofcriticalpointsisfinite. Proposition. K is an (2n 1)-manifold for all " > 0suciently small. Proof: Since ⌃(V )isfinite,weknowthatCn+1 ⌃(V )isanopen(2n +2)-manifold. Now \ 0 C is a regular value for 2 : n+1 f Cn+1 ⌃(V ) C ⌃(V ) C | \ \ ! and so V ⌃(V )isasmooth2n-manifold, by the preimage theorem. Now consider the radius\ map 2 r : V ⌃(V ) R,z z z0 . \ ! 7!k k Since r is a polynomial, it has finitely many critical points (as above) and so finitely many critical values. Hence we can find "2 > 0smallerthanallcriticalvaluesin R (except possibly 0). The preimage theorem then gives that

1 2 r (" )=(V ⌃(V )) S \ \ " is a smooth (2n 1)-manifold. Since the critical points of f are isolated, we can pick " sucently small so that S ⌃(V )= , even if z is itself a critical point. " \ ; 0 32 Remark. This also works in the real case and the case of more general algebraic varieties.

Inclusion i : K , S" in then a codimension 2 embedding. If K was homeomorphic 2n 1 ! to S , then this would be a higher dimensional knot. As we shall soon see, this K often has an exotic di↵erentiable structure and invariants that distinguish di↵erentiable structure are closely related to invariants for knots. We start with the less interesting case when z V is a regular point. 0 2 2 Example. If z0 V is a regular point of f, then z0 V ⌃(V )andsor : z z z0 2 2 \ @2f 7!k k now has a critical point at z0. In fact, z0 is non-degenerate since @z @z =2ij. i j p Since ⌃(V )isfiniteanddoesnotcontainz , we can pick " suciently small so that 0 B" ⌃(V )= , i.e. so that r : V B" R. Now r is a Morse function mapping from a compact\ 2n-manifold; so, by Morse’s\ lemma! and the fact that r(z) 0, we can find local real coordinates u , ,u for V B near z such that 1 ··· 2n \ " 0 r(u , ,u )=u2 + + u2 . 1 ··· 2n 1 ··· 2n Rechoosing " so that V B" corresponds to this neighbourhood, we now have a di↵eo- morphism \

2n 2 2 2 2n 1 K = V S" (u1, ,u2n) R : u1 + + u2n = " = S . \ ! { ··· 2 ··· } 2n 1 2n+1 As mentioned before, K , S" is an embedding S , S and so is a higher dimensional knot. However,! K is not knotted. Indeed,! the above argument can be n+1 extended so that r : C B" R is given local real coordinates u1, ,u2n+2 and so that V B is retrieved by\ the slice! coodinates u = u =0. ··· \ " 2n+1 2n+2 2n 1 So if z0 is a regular point, the corresponding K is both di↵eomorphic to S and is unknotted when embedded into S2n+1. In search of more interesting examples, we now consider two cases where z0 is a critical point. 2 3 Example. Let f(z1,z2)=z1 + z2, which has a single critical point at the origin. Since K is a smooth 1-manifold, we know that K is di↵eomorphic to S1. However, by finding K explicitly, it is possible to show that K , S3 is knotted. ! 2 3 2 2 2 Pick " > 0andlet(z1,z2) K = V (f) S" so that z1 = z2 and z1 + z2 = " . Combining gives that 2 \ | | | | z 3 + z 2 = "2. | 2| | 2| Since the left side is strictly increasing, there is a unique ⌘ > 0suchthat z2 = ⌘ (by, for example, the intermediate value theorem) and so also a unique ⇠ > 0suchthat| | z = ⇠. | 1| Hence (z1,z2)liesinatorus

2 T = (z1,z2) C : z1 = ⇠, z2 = ⌘ . { 2 | | | | } 3i✓ ki⇡/3 2/3 2i✓+ki⇡/3 If z1 = ⇠e , then z2 = ⌘e z1 = ⌘e for some 1 k 3correspondingtothe three roots of unity. By swapping ✓ for ✓ +2⇡/3successivelychangesthevalueof  k, so we can assume k = 3. Hence K = (⇠e3i✓, ⌘e2i✓) T : 0 ✓ < 2⇡ . { 2  } which is the (2, 3) torus knot, i.e. the right-handed trefoil knot. It can easily be shown, using the standard tools of knot theory, that this is in fact knotted. Note that this always works for 2 and 3 replaced by any coprime p and q,givinga(p, q)torusknot.

33 Figure 4: The (2, 3) torus knot, i.e. the right-handed trefoil.

To have any hope of finding an exotic sphere, we must of course consider higher-dimensional varieties. In particular, the above example can be generalised by considering

f(z , ,z )=z2 + + z2 + z3 . 1 ··· n+1 1 ··· n n+1 Here K , S2n+1 turns out to give what is refered to as a generalised trefoil knot. As ! 2n 1 we shall soon see, K is homeomorphic to S when n is odd. When n =1(above)or 2n 1 n =3,thisisdi↵eomorphic to S however when n =5,wecanshowthatthisisan exotic 9-sphere. The focus of the rest of this section will be to outline to tools needed to establish this fact. For the sake of introducing a piece of standard terminology, we note that this is an example of a Brieskorn variety, which are those links K around the origin which come from polynomials of the form za1 + + zan+1 for integers a 2. 1 ··· n+1 i 6.1 The Alexander Polynomial of K We now construct an invariant for K which generalises the Alexander polynomial for knots. For a modern construction of the Alexander polynomial for knots, see [21]. 1 As in our discussion above, let deg f>1, V = f (0) and z V be a critical point 0 2 (which we showed must be isolated). Let S" be a sphere centred at z0 with " suciently small so that K = V S is a smooth (2n 1)-manifold. \ " Consider the knot complement S K = z S : f(z) =0 and the well-defined map " \ { 2 " 6 } : S K S1,z f(z)/ f(z) . " \ ! 7! 1 i✓ Let F = (e )= z S K : arg(f(z)) = ✓ be the fibres of this map. ✓ { 2 " \ } We will assume the following fact, for which the reader is directed to [20] for a proof. Theorem (Milnor’s Fibration Theorem). The radial projection map induces a smooth fibre bundle F S K S1. ✓ " \ Each fibre F✓ is a smooth 2n-manifold which is homotopic to the wedge of µ 1copies of Sn and has F = F K, where the common boundary K is (n 2)-connected. ✓ ✓ [

34 1 1 To define our invariant ,wefirstdefineanactionof⇡1(S )onthefibreF0 = (1). If z F and ⇡ (S1) is a loop based at 1, then the fact that above is a fibre bundle 2 0 2 1 and hence a fibration (see [13]) means we can lift to a unique path : [0, 2⇡] S" K starting at z. Then action15 is then defined as : z (2⇡). ! \ 7! 1 Let h : F0 F0 denote the action of the generator id ⇡1(S )onF0e, which induces an automorphism! 2e h : H (F0) H (F0). ⇤ ⇤ ! ⇤ µ Now note that the above theorem gives that Hi(F✓)=Z when i = n and is trivial for all µ µ other 1 i 2n, so the only non-trivial map is hn : Z Z . Our invariant then comes from the characteristic polynomial of this linear map. ! Definition (Alexander polynomial). The Alexander polynomial of a link K is (t)=(t)=det(tI h ), K n n where I is the identity map on F0.

Since hn is an isomorphism, we must have that det(hn)= 1. Therefore (t) is a polynomial with integer coecients of the form ±

m m 1 t + a1t + + am 1t 1. ··· ± To take things further, we note that the monodromy action of id ⇡ (S1)inducesa 2 1 mapping F0 [0, 2⇡] S" K given by (z,t) (t) Ft. We can then check this gives an isomorphism⇥ ! \ 7! 2 H (F [0, 2⇡],F 0 F e2⇡ ) H (S K, F ). i 0 ⇥ 0 ⇥ { } [ 0 ⇥ { } ! i " \ 0 Furthermore, the long exact sequence for relative homology can be used to show that the group on the left is Hi 1(F0). Replacing Hi(S" K, F0)withHi 1(F0)inthelongexact sequence for the pair (S K, F )thengivesusthefollowing.\ " \ 0 1 Lemma (Wang sequence). The fibre bundle F0 , S" K S induces a long exact sequence ! \ !

h I H (S K) H (F ) ⇤ ⇤ H (F ) H (S K) ··· i+1 " \ i 0 i 0 i " \ ··· Since K is an (n 2)-connected (2n 1)-manifold and bounds the (n 1)-connected 2n-manifold F0, we can follow a similar approach to the one outlined at the end of the 2n 1 section on cobordism to characterise when K is homeomorphic to S .

Indeed, since S" K is a compact subspace of a (2n + 1)-sphere, Alexander duality (see \ 2n i 2n i [13]) gives that Hi(S" K) = H (K)andH (K) = Hi 1(K)byPoincar´eduality. ⇠ ⇠ The Wang sequence is\ then

hn In 0 Hn(K) Hn(F0) Hn(F0) Hn 1(K) 0. We know from our comments of highly-connected manifolds that K is homeomorphic to 2n 1 S i↵ Hn 1(K)=0i↵ the map hn In is an isomorphism which happens precicely when (1) = det(h I )= 1. Hence we have shown the following. n n ± 15This is the same construction we have already seen in covering space theory, sometimes known as the monodromy action. Also note that this corresponds to the construction of the infinite cyclic cover found in [21], and so indeed can be seen to generalise the Alexander polynomial.

35 2n 1 Theorem. If n 3, then K is homeomorphic to S i↵(1) = det(I h )= 1. n n ± 2n 1 Now assume K is homeomorphic to S . To characterise the smooth structure, we can of course use the characteristic class invariants we have already established. However, complete invariants can also be derived. There are two distinct cases. 1. When n is odd, i.e. dim K 1mod4,thesmoothstructureiscompletelydeter- ⌘ mined by the Kervaire invariant c(F0) Z/2Z (a definition can be found in [4]). 2 2. When n is even, i.e. dim K 3mod4,thesmoothstructureiscompletelydeter- mined by the signature of the⌘ intersection product

s : Hn(F0) Hn(F0) Z. ⇥ ! Whilst proofs of these facts are beyond the scope of this essay, we will state a remarkable result of Levine (see [11]). Namely, if n is odd, then

0, if ( 1) 1mod8 c(F0)= ⌘ ± 1, if ( 1) 3mod8 ( ⌘ ± Hence to verify that a particular K of dimension (4n +1)isanexoticsphereamountsto computing (t). 6.2 Examples of Exotic Spheres Here we will present two constructions of exotic spheres and illustrate the two approachs mentioned above for classifying their smooth structure. In both cases, we must compute the Alexander polynomial (or at least the sum of its coecients). Here is where it is helpful to restrict to the case of Brieskorn varieties, where the Alexander polynomials can be classified as follows.

Theorem (Brieskorn-Pham). Let F0 be the fibre associated with the Brieskorn variety za1 + + zan+1 . Then H (F )hasrankµ =(a 1) (a 1) and 1 ··· n+1 n 0 1 ··· n+1 (t)= (t ! ! ), 1 ··· n+1 Y where the product is over all the non-trivial ai-th roots of unity wi for each i. For a proof, the reader is directed to [20]. We now consider the two examples. 1. Consider, as mentioned before, the (2n 1)-dimensional link K corresponding to f(z , ,z )=z2 + + z2 + z3 . 1 ··· n+1 1 ··· n n+1 1 To compute (t), note that w1 = = wn = 1andwn+1 = µ± , where µ is a non-trivial third root of unity. Hence···

n n 1 2 n 1 2 n (t)=(t ( 1) µ)(t ( 1) µ )=t ( 1) (µ + µ )t +1=t +( 1) t +1, which gives that (1) = 2 + ( 1)n and ( 1) = 2 + ( 1)n+1. If n =2m +1isoddandn 3, this shows that K is homeomorphic to S4m+1.We can also compute that ( 1) = 3 and so

c(F0)=1.

36 It remains to detemine which value of c(F0)correspondstothestandardspherein each dimension. When n = 3, we must have the standard smooth structure since there are no exotic 5-spheres. However, when n =4,thisturnsouttogiveanexotic 9-sphere. In fact, this is the Kervaire 9-sphere we considered in the previous section. 2. Consider the 7-dimensional link K corresponding to

2 2 2 5 3 f(z1,z2,z3,z4,z5)=z1 + z2 + z3 + z4 + z5.

1 2 1 Now (t)hasw1 = w2 = w3 = 1, w4 = ⌫± , ⌫± and w5 = µ± , where ⌫ is a non-trivial fifth root of unity and µis as above. We compute (1) directly:

i 1 i i 2i (1) = (1 + µ⌫ )(1 + µ ⌫ )= (1 ⌫ + ⌫ ). i= 1, 2 i= 1, 2 Y± ± Y± ± 2 2 1 Now note that (1 ⌫ + ⌫ )(1 ⌫ + ⌫ )canbeexpandedouttogive 2 1 2 2 1 2 (1 ⌫ + ⌫ )+( ⌫ + ⌫ 1) + (⌫ ⌫ + ⌫)=⌫ 1 2 2 2 1 and so (1 ⌫ + ⌫ )(1 ⌫ + ⌫)=⌫ , by replacing ⌫ with ⌫ in the formula. 2 2 7 Hence (1) = ⌫ ⌫ =1andsoK is homeomorphic to S . · We can show this is an exotic 7-sphere by showing that the signature of the bounding manifold N = F0 is (N)= 8, and then computing the Pontryagin classes using the methods we have used throughout this essay. Generalising the last example, Hirzebruch established that the link around the origin of

2 2 6k 1 3 z1 + + z2n 1 + z2n + z2n+1 ··· 4n 1 was homeomorphic to S and characterised the k for which this gives an exotic sphere. For example, when n =2thisgivesdistinctexotic7-spheresfork =1,...,28. As we shall see in the following section, these are all the possible exotic 7-spheres. We conclude by mentioning a generalisation of this result, proven by Brieskorn: Theorem (Brieskorn). Every homotopy (2n 1)-sphere for n 4thatboundsamanifold 1 with trivial tangent bundle is the link K around the origin of a hypersurface V = f (0) where f(z , ,z )=za1 + + zan+1 , 1 ··· n+1 1 ··· n+1 for some integers a 2. i We will return to these results in the following section, showing how they fit in amongst the bigger picture of the classification.

37 7 Classification of Exotic Spheres

The goal of this section will be to bring together many of the idea presented in this essay and to talk about the classification of exotic spheres that exist in any given dimension. For n 5, it was reduced to solving classical problems in Algebraic Topology by Milnor and Kervaire in 1963 (see [4]). We will give an overview of the methods used to do this and, finally, will make a few brief remarks about how the various constructions we have presented fit in to this picture. The idea is to find a way to put an algebraic structure on the di↵eomorphism classes of manifolds homeomorphic to Sn. In particular, the operation we will use is connected sum M#N. This can be defined in a smooth setting and we can check that it is unique up to orientation-preserving di↵eomorphisms. n Let ⇥n be the set of oriented n-manifolds homotopic to S up to h-cobordism. If n 5, we know that being h-cobordant is the same as being di↵eomorphic and being homotopic n n to S is then same as being homeomorphic to S . Thus ⇥n also corresponds to the set of distinct smooth structures on Sn.

Proposition. ⇥n is an abelian group under connected sum. n n Proof: First note that if M ⇥n, then S #M is di↵eomorphic to M, so [S ] acts as the identity. That # is associative2 and commutative is immediate. To show closure, simply note that the connected sum of two copies of Sn is a homotopy sphere.

Finally, let M ⇥n be a homotopy n-sphere. To show that inverses exist, we claim that M#( M2)=Sn. To prove this, note that M#( M) bounds the cylinder B =(M Dn) [0, 1] for a small n-disc Dn M. Since Bis a compact contractible manifold\ with⇥ simply-connected boundary,✓ it must be di↵eomorphic to Dn by our first corollary of the h-cobordism theorem. In particular M#( M)=Sn, which completes the proof. Thus the problem of finding the number of exotic n-spheres is reduced to finding the order of the group of h-cobordism classes of homotopy n-spheres.

It can be shown that ⇥n is finite for every n 5. The proof of this fact, which we will not give, relies on identifying a finite-index subgroup and then showing that the subgroup is finite. Definition. A smooth manifold M is said to be parallelisable if its tangent bundle is trivial and almost-parallelisable if M F is parallelisable for some finite set F . \ We write bPn+1 for the set of h-cobordism classes of n-manifolds which are boundaries of parallelisable manifolds. Note that bP is a subgroup of ⇥ since, if M ,M bP bound parallelisable mani- n+1 n 1 2 2 n+1 folds B1 and B2 respectively, then M1#M2 bounds the parallelisable manifold formed by attaching B1 and B2 using an approriate operation. We will now briefly review a few basic facts and definitions from homotopy theory. Firstly, note that the suspension homomorphism induces a map

⇡ (Sk) ⇡ (S(Sk)) = ⇡ (Sk+1) n+k ! n+k+1 n+k+1 and this map is an isomorphism for n>k+1. Wecan thus define the nth stable homotopy s k group of spheres to be the group this stabilises to, i.e. ⇡n =limk ⇡n+k(S ). Some !1 38 examples of these groups can be found in the table below. By noting that the homotopy groups of spheres are finite except for a class of known examples, it can be shown that s the ⇡n are finite for all n. k Secondly, note that a homomorphism Jn : ⇡n(SO(k)) ⇡n+k(S )canbeconstructedas n !k 1 k 1 follows. An element of ⇡n(SO(k)) has the form S S S and so, by the Hopf construction (see [13]), this induces a map ⇥ !

n+k n k 1 k 1 k S = S S S(S )=S , ⇤ ! k where denotes the join of two spaces, and so gives an element of ⇡n+k(S ). This in ⇤ s turn induces a map J : ⇡n(SO) ⇡n, where SO denotes the direct limit of the special orthogonal groups, known as the!J-homomorphism. It can be shown that the image, s which we denote by J, is a cyclic subgroup of ⇡n. These are related to our groups as follows. Theorem. There is an injective homomorphism ⇥ /bP ⇡s /J. n n+1 ! n s In particular, since ⇡n is finite, this shows that ⇥n/bPn+1 is also finite. This relationship can be seen in the examples tabulated below.

n 12 3456 7 8 9 10 s ⇡n Z2 Z2 Z24 00Z2 Z240 Z2 Z2 Z2 Z2 Z2 Z6 s ⇡n/J 0 Z2 000Z2 0 Z2 Z2 Z2 Z6 ⇥n/bPn+1 00 0000 0 Z2 Z2 Z2 Z6 Interestingly, this map turns out to either be an isomorphism of have index 2. The above table shows this it is index 2 when n =2or6andisanisomorphismotherwise.Itcan be shown that it is always an isomorphism when n 3mod4. ⌘ Remark. Classifying the cases for which the index is 2 is equivalent to the Kervaire invariant problem which we discussed at the end of the previous section. Recall that recent results have deduced that this is only the case in dimensions n =2, 6, 14, 30, 62 and possibly 162.

We now review a few basic facts about bPn+1. Firstly, it is finite cyclic group which is trivial when n is even and has order 1 or 2 when n 1mod4.Thecomplicatedcaseis when n =4m 1, which we construct below. ⌘ Let m be the smallest positive signature obtained by an almost-parallelizable 4m-manifold without boundary. If M bP bounds the parallelisable 4m-manifold B, then let 2 4m M (B)mod . 7! m

This can be shown to be a well-defined homomorphism bP4m Zm using the facts we previously established about the signature. It can also be shown! to be injective and have image of size m/8, hence showing that bP4m is cyclic. Furthermore, we have the following result.

Theorem. Let B2m be the Bernoulli numbers, jm = J(⇡4m 1(SO)) and am is 1 when m is even and 2 when m is odd. Then | |

2m 1 2m 1 B2m amjm =2 (2 1)| | . m m 39 Hence, since the map ⇥ /bP ⇡s /J is an isomorphism for n =4m 1, we get that n n+1 ! n s s 2m 4 2m 1 B2m ⇡4m 1 am ⇥4m 1 = ⇡n/J m/8=2 (2 1)| || | . | | | |· m

Subject to computing whether or not bPn+1 has order 1 or 2 when n 1 mod 4, we can extends our above table to get: ⌘

n 123456 7 8910111213141516 ⇥n 11111128286 9921 3 2 162562 bP| | 11111128121 9921 1 1 81281 | n+1|

Note that ⇥n only gives us the number of exotic n-spheres for n 5. In particular, this shows that we found 4 of the 28 exotic 7-spheres and 64 of the 16256 exotic 15-spheres by the sphere bundles construction. Of course, in many cases we can also determine ⇥n as a group from this information alone. For instance, whenever ⇥ = bP ,weknowthat | n| | n+1| ⇥n is cyclic. So the 28 exotic 7-spheres form a cyclic group under connected sum. We now consider how this classification relates to the constructions from the last two sections.

Firstly, recall the Milnor spheres ⌃M of dimensions 4n 1andtheKervairespheres⌃K of dimensions 4n +1thatweconstructedinthesectiononPlumbing.Wecaneasilyverify that they both bound parallelisable manifolds. In fact, it can be shown that they each generate bP4n and bP4n+2 respectively. The latter case shows the equivalence of the two forms of the Kervaire invariant problem that we have mentioned in this essay. We might also ask how the exotic spheres we found as Brieskorn varieties fit into this picture. Recall that the sphere K we constructed was the boundary of the fibre F0. It can be shown that F0 is always parallelisable and so each K lies in bPn+1. Hence, since bP10 = 2, the exotic 9-sphere we found was the only bounded parallelisable sphere other| than|S9.Thisprovesthatitisdi↵eomorphic to the corresponding Kervaire sphere. Furthermore, the Brieskorn varieties which come from polynomials of the form

2 2 6k 1 3 z1 + + z2n 1 + z2n + z2n+1 ··· n are homotopy (4n 1)-spheres and can be shown to represent the class of ( 1) k bP4n (where we take 1 to be the generator). For example, when n =2,theexotic7-spheres 2 are all bounded parallelisable and so are precisely these varieties for k =1,...,28, as mentioned in the previous section. More generally, recall the result proven by Brieskorn that said that every homotopy (2n 1)-sphere for n 4thatboundsaparallelisable manifold corresponds to a Brieskorn variety. This shows the Brieskorn varieties generate bP for every n 4. We now conclude by making a number of final remarks. 2n Firstly, and perhaps rather unexpectedly, we comment on the real world applications of exotic spheres. Recall that spacetime can be meaningfully modelled as a smooth manifold in such a way that changes in reference frames corresponds to di↵eomorphisms of space- time. We therefore would expect exotic spheres to have at least some physical significance, and in fact many believe they do. In particular it is conjectured by Edward Witten that gravitational instantons and/or solitons take the form of very exotic spheres, i.e. those that do not bound parallelisable manifolds. Such ideas can be found in a paper written by Witten in 1985 (see [12]) where he argues that the existence of exotic spheres are needed to explain certain global gravitiational anomolies. For a more recent exposition, see [22].

40 Secondly, we may wonder whether or not the results we have proven about smooth (i.e. k C1) structures would be completely di↵erent had we considered C structures for k 1. However this is not the case by the delightful result that any Ck structure can be extended to a C1 structure uniquely up to di↵eomorphisms. In fact, the corresponding map from k C structures to C1 structures is a bijection on the corresponding equivalence classes. k Hence the classification of C structures are completely identical for any k Z 0 . 2 [ {1} Finally we remark that these ideas developed in a number of di↵erent directions once the h-cobordism theorem was proven and the classification of exotic spheres was completed. On one hand, note that the 4th dimension which remains characteristically absent from any of the results we have established: it is still not known whether or not there exists an exotic 4-sphere. In fact, it is not even known whether or not the number of smooth structures is finite. The 4th dimension tends to be a wild and unusual place; for example, it can be shown that Rn has a unique smooth structure for all n =4,thoughinfinitelymany when n = 4. Alternatively, one might consider leaving spheres6 and perhaps even simply- connected spaces altogether. In this direction, we note that the h-cobordism theorem was later generalised to the s-cobordism theorem which states that a (general) h-cobordism X between N and M is trivial if and only if an invariant ⌧(X, M)knownastheWhitehead torsion vanishes. These ideas live on today in the field of Algebraic K-Theory, where many of Milnor’s next great accomplishments can be found.

41 8 References

[1] J. Milnor, On manifolds homeomorphic to the 7-sphere, Ann. of Math. (2), Vol. 64, p399-405 (1956) [2] J. Milnor, Differentiable Structures on Spheres, American Journal of Mathematics (4), Vol. 81, p962-972 (1959) [3] J. Milnor, Differentiable Manifolds which are Homotopy Spheres, (mimeographed), Princeton University (1959) [4] M. Kervaire, J. Milnor, Groups of homotopy spheres I, Ann. of Math. (2), Vol. 77, p504-537 (1963) [5] M. Kervaire, A manifold which does not admit any differentiable structure, Commen- tarii Mathematici Helvetici (1), Vol. 34, p257-270 (1960) [6] N. E. Steenrod, The Classification of Sphere Bundles, Ann. of Math. (2), Vol. 45, p294-311 (1944) [7] N. Shimada, Differentiable structures on the 15-sphere and Pontrjagin classes of cer- tain manifolds, Nagoya Math. J., Vol. 12, p59-69 (1957) [8] C. E. Dur´an, Pointed Wiedersehen Metrics on Exotic Spheres and Diffeomorphisms of S6, Geometriae Dedicata (1), Vol. 88, p199-210 (2001) [9] M. Joachim, D. J. Wraith, Exotic Spheres and Curvature, Bulletin of the American Mathematical Society (4), Vol. 45, p595-616 (2008) [10] C. T. C. Wall, Classification of (n 1)-connected 2n-manifolds, Ann. of Math, Vol. 75, p163-189 (1962) − [11] J. Levine, Polynomial invariants of knots of codimension two, Ann. of Math. (2), Vol. 83 p537-554 (1966) [12] E. Witten, Global Gravitational Anomalies, Commun. Math. Phys. Vol. 100, p197- 229 (1985) [13] A. Hatcher, Algebraic Topology, Cambridge University Press (2001) [14] N. E. Steenrod, The Topology of Fibre Bundles, Princeton University Press (1951) [15] J. Milnor, J. Stasheff, Characteristic classes, Ann. of Math. Studies, Vol. 76 (1974) [16] J. Milnor, Morse Theory, Princeton University Press (1963) [17] R. E. Strong, Notes on Cobordism Theory, Princeton University Press (1968) [18] A. A. Kosinski, Differential Manifolds, Academic Press Inc. (1993) [19] W. Browder, Surgery on Simply-Connected Manifolds, Springer (1972) [20] J. Milnor, Singular Points of Complex Hypersurfaces, Ann. of Math. Studies, Vol. 61 (1968) [21] W. B. R. Lickorish, An Introduction to Knot Theory, Springer (1997) [22] R. A. Baahio, Quantum Topology and Global Anomalies, Advanced Series in Mathe- matical Physics, Vol. 23 (1996)

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